Shape Reconstruction Incorporating Multiple Non-linear Geometric Constraints
Naoufel Werghi,Robert Fisher,Anthony Ashbrook and Craig Robertson Division of Informatics,University of Edinburgh
5Forrest Hill,Edinburgh EH12QL,UK
Email:{naoufelw,rbf,anthonya,craigr}@https://www.doczj.com/doc/bb16317447.html,
Abstract.This paper deals with the reconstruction of3D geometric shapes based on observed noisy3D measurements and multiple coupled non-linear shape constraints.Here a shape could be a complete object,a portion of an object,a part of a building etc.The paper suggests a general incremental framework whereby constraints can be added and integrated in the model reconstruction process,resulting in an optimal trade-off between minimization of the shape ?tting error and the constraint tolerances.After de?ning sets of main constraints for objects containing planar and quadric surfaces,the paper shows that our scheme is well behaved and the approach is valid through application on different real parts.This work is the?rst to give such a large framework for the integration of numerical geometric relationships in object modelling from range data.The technique is expected to have a great impact in reverse engineering applications and manufactured object modelling where the majority of parts are designed with intended feature relationships.
Keywords:Reverse engineering,geometric constraints,constrained shape reconstruction, shape optimization
Abbreviations:CAD–Computer Aided-design;3D–Three-Dimensional;LS–Least squares
Table of Contents
1Introduction2 2Related work4 3The geometric constraints6 4Optimization of shape satisfying the constraints8 5Implementation14 6A simple example14 7Experiments17 8Conclusion31
c2000Kluwer Academic Publishers.Printed in the Netherlands.
2
3D scanning
CAD model
Design with CAD
Optical measurement Optimization
Rapid prototyping
1
2
3
4
Figure 1.The production-perfection cycle of a part
1.Introduction
The framework of this work is reverse engineering.In parts manufacturing,reverse engineering is typically concerned with measuring an existing object so that a surface or solid model can be deduced in order to take advantage of CAD/CAM technologies.It is also often necessary to produce a copy of a part when no original drawings or documentation are available.In other cases we may want to re-engineer an existing part,when analysis and modi?cations are required to construct a new improved product.Even though it is possible to turn to a computer-aided design to fashion a new part,it is only after the real model is made and evaluated that we can see if the object ?ts the real world.For this reason designers rely on real 3D objects (real scale wood,clay models)as starting point.Such a procedure is particularly important to areas involving aesthetic design e.g.the automobile industry or generation of custom ?ts to human surfaces such as helmets,space suits or prostheses.For these reasons reverse engineering is a fundamental step of the now-standard production-perfection cycle of part (Figure.1).This process starts with the CAD stage.Next (step 2),the rapid prototyping stage converts the CAD data into a real prototype.Rapid prototyping is a technique allowing the direct production of prototypes by a computer-controlled process.Often,the shape of the produced object undergoes some improvement carried out by hand to adapt it to its real environment (step 3).The hand-improved model is
3
3D scanning
CAD model
Design with CAD
Optical measurement Optimization
Rapid prototyping
New constraints
1
2
3
4
Figure 2.Many hand-worked optimization (step 3)could be replaced by establishing new constraints on the shape and incorporating them in the model design process.
back again into the digital world of CAD through 3D optical measurement techniques (step 4),for instance a 3D laser scanner.
In this process the notion of constraints is normally involved in step 1where geometric relationships between object features together with 3D measurement data contribute in the production of the optimal object model shape.
The ?rst motivation behind incorporating geometric constraints is that models needed by industry are generally designed with intended geometric relationships between the object features so this aspect should be exploited rather than ignored.The consideration of these relationships is actually neces-sary because some attributes of the object would have no sense if the object modelling scheme did not take into account these constraints.For example,take the case when we want to estimate the distance between two parallel planes:if the plane ?tting results gave two planes which are not parallel,then the distance measured between them would have no signi?cance.Fur-thermore exploiting the available known relationships would be useful for reducing the effects of registration errors and mis-calibration,thus improving the accuracy of the estimated part features’parameters and consequently the quality of the modelling.
The second motivation is that once the part is produced (step 2)many improvements are carried manually (step 3)to optimize the part and make it ?t with the real world (e.g ?t with another part,adjust the part to ?t a particu-
4
lar customer).These improvements could be represented by new constraints
on the part’s shape.By integrating these constraints into the CAD design
process step(Figure.2)the work piece optimization would be reduced to the
minimum tasks and hence many cycles in the part production process would
be saved.In other cases,such improvements could not be achieved by hand
due to the complexity of the object or when we want to extend the application
of the process to complex environments such as buildings or industrial plants.
Our problem is presented as follows:Given sets of3D measurement points representing surfaces belonging to a certain object,we want to estimate
the different parameters of the surfaces,taking into account the geometric
relationships between these surfaces and the speci?c shapes of surfaces as
well.
A state vector p is associated to the object,which includes all paramet-
ers related to the patches.The shape de?ned by the parameter vector p has
to best?t the data while satisfying the constraints.Consider F p to be an
objective function de?ning the relationship between the set of data and the
parameters and C k p,k1M the set of constraint functions de?ning the geometric constraints.C k p is a vector function associated with constraint k.The problem can be then stated as follows:Find the parameter vector p
minimizing the function F p subject to the constraints
C k pτk k1M(1) Hereτk represents the tolerance related to the constraint C k.Ideally the tol-erances have zero values,but practically,for geometric constraints they are assigned certain values which re?ect the allowed geometric inaccuracies in the relative locations and shapes of features.It is up to the designer to set the tolerances,however an appropriate de?nition of the tolerances for a given object can be set up by using the scheme developed by Requicha[16].
As a simple example consider the three surfaces of a tetrahedron(Fig.3).
The surfaces have three orientation constraints re?ecting the three angles
900,900and1200between the three surface normals.Consider p a vector
containing the parameters of the surfaces,p has then to?t the data points as-
sociated with the surfaces,minimizing a least squares error function and also
satisfying the three constraint functions associated to the surface orientations.
2.Related work
A review of the main reverse engineering research in the CAD community
[7,18,19,22]revealed that the exploitation of geometric constraints has not
been fully investigated.This lack was discussed in the survey work of Varady
et al[20].
5
S1
S2
S3
Tetrahedron
Figure3.The tetrahedron object with the extracted surfaces
Incorporating geometric relationships in object modelling has to tackle two problems.The?rst is how to represent the constraints.The second is how to integrate these constraints into the shape?tting process.These two aspects are not entirely independent,the shape?tting technique imposes restrictions on the constraint representation and vice versa.
A?rst step in the direction of incorporating constraints for ensuring the consistency of the reconstruction was done by Porrill[15].He linearized a set of nonlinear constraints and combined them with a Kalman?lter applied to wire frame model construction.Porrill’s method takes advantage of the recursive linear estimation of the Kalman?lter,but guarantees satisfaction of the constraints only to linearized?rst order.Additional iterations are needed at each step if more accuracy is required.This last condition has been taken into account in the work of De Geeter et al[4]by de?ning a“Smoothly Constrained Kalman Filter”.The key idea of their approach is to replace non-linear constraints by a set of linear constraints applied iteratively and updated by new measurements in order to reduce the linearization error.However,the characteristics of Kalman?ltering makes these methods essentially adapted for iteratively acquired data and many data samples.Moreover,there was no mechanism for determining how successfully the constraints were satis?ed and only lines and planes were considered in both of the above works.
The constraints considered by Bolle et al[2]in their approach to3D object position covered only the shape of the surfaces.They chose a speci?c representation for the treated features:plane,cylinder and sphere.
Compared to Porrill’s and De Geeter’s work,our approach avoids the drawbacks of linearization,since the constraints are completely implemented. Moreover,our approach covers a larger category of feature shapes.Regard-ing the work of Bolle[2],the type of constraints which can be held by our approach go beyond the restricted set of surface shapes and cover also the geometric relationships between object features.To our knowledge the work appears to be the?rst to give such a large framework for the integration of geometric relationships for object reconstruction.
6
3.The geometric constraints
The set of constraints associated with a given object can be divided mainly into two categories.The?rst one is the surface intrinsic constraints covering the geometric properties which arise from the speci?c shapes of the surfaces. This category includes particular properties of the surface such as symmetry with respect to a point or a line.For quadric surfaces such as cones or cross-section cylinders this property is the circular shape of the surface.
The second category named,the feature extrinsic constraints,de?nes the geometric and topological relationships between the different object features. Table I summarizes these relationships.We notice here that points and lines in this table may be either physical features of the object like summits or vertices and edges or implicit features like centres,axes of symmetry.This list is not exhaustive and this classi?cation may not be unique.Nevertheless it covers a large number of constraints in manufactured objects.
Table I.Relationships between features.
line quadric surface
coincident inclusion
separation separation
line coincident inclusion
relative orientation relative orientation
separation separation
-coincident
relative orientation
separation
quadric surface-coincident
relative orientation
separation
3.1.C OINCIDENCE CONSTRAINTS
Shapes commonly contain features which are associated to the same geomet-ric entity(Figure.4.a)or which coincide at the same position(Figure.4.b).In the?rst case these constraints are implicitly imposed by considering the same parameters for each feature.In the second case the parameters associated to each feature are equated and the resulting equations have then to be satis?ed.
7
E E P 11
2
2
P +
Cyl Cyl Cir Cir C
D
2
1
12
(a)
(b)
Figure 4.(a):The two edges E 1and E 2belong to the same line.The two faces P 1and P 2are
associated to the same plane.(b)The centres of the circles Cir 1and Cir 2coincide at the same point C .The cylinders Cyl 1and Cyl 2have a common axis.
3.2.I NCLUSION
CONSTRAINTS
A particular feature point may be included in an object feature e.g.line,plane or quadric patch.Similarly a feature line may be included in a plane or a particular quadric surface (Fig.5)such as a cylinder and a cone.3.3.R ELATIVE
ORIENTATION CONSTRAINT
There are many orientation relationships which can be deduced and exploited in a given part,such as the two common particular cases of parallelism and orthogonality (Fig.6.a).The presence of these two characteristics is easily detected in an object.
P
axecyl
Cyl
Cyl
E
(a)
(b)
Figure 5.(a):The axis of the cylinder patch Cyl is included in the plane P .(b)The line associated to the edge E is included in the cylinder Cyl .
8
P P P Cyl
1
2
3
P P 1
d
2
P P P P P P 1
2
3
4
5
6
(a)
(b)
(c)
d
Figure 6.(a):Each pair of planes P 1P 2P 3makes an angle of 90o ,the axis of the cylinder Cyl is orthogonal to P 1.(b):The planes P 1P 2are separated by distance d .(c):Each pair of parallel planes of the hexagonal prism are separated by the same distance.
3.4.R ELATIVE
SEPARATION CONSTRAINT
The relative separation between features can be exploited when the distance between parallel features (Fig.6.b)is already known or needs to be imposed or when the object has a symmetry aspect leading to some separation distance relationships (Fig.6.c).3.5.O THER
CONSTRAINTS
There are also other types of constraints like those imposed directly on the surface parameters as a consequence of the surface representation e.g.the representation of a plane by the equation ax by cz d 0where a b c is normal vector to the plane and d is the distance of the plane to the origin requires that the sum of the squared elements of the normal to be equal to one.Such constraints are called the unit constraints.
4.Optimization of shape satisfying the constraints
Given sets of 3D measurement points representing surfaces belonging to a certain object,we want to estimate the different surface parameters,tak-ing into account the geometric relationships between these surfaces and the speci?c shapes of surfaces as well.
A state vector p is associated to the object,which includes all parameters related to the different patches.The vector p has to best ?t the data while satisfying the constraints.Consider F p to be an objective function de?n-ing the relationship between the measured data points and the parameters.
9 This function is generally a minimization criterion(e.g.sum of least squares residuals,maximum likelihood function,etc.).
Consider C k p,k1M,the set of constraint functions de?ning the geometric constraints where C k p is a vector function associated with constraint k.The problem can be then stated as follows:
minimize F p
subject to the constraints C k pτk k1M(2) Thus the problem which we are dealing with is a constrained optimiza-tion problem.
4.1.T HE OBJECTIVE FUNCTION
Consider S1S N the set of surfaces and p1p N the set of parameter vectors related to them.Each vector p i has to minimize a given surface?t error cri-terion J i associated with the surface S i such as the least squares error criterion. The set of the parameter vectors has then to minimize the following object function:
J J1J2J N(3) By considering a polynomial description of the surfaces,each surface S i can be represented by:
h i T p i0(4) where h i is the measurement vector with each component of the form xαyβzγfor someαβγ.For instance a plane surface de?ned by the equation ax by cz d0,has the measurement vector is h x y z1T.For a sphere de?ned by a x2y2z22ux2vy2wz d0,it is h x2y2 z2x y z1
This formulation has the advantage to lead to a compact quadric ex-pression of the objective function because of its linearity with respect to the parameters.Indeed,given m i measurements,the least squares criterion related to the equation(4)is
J i
m i
∑
l1
h i
l
T
p i2p i T H i p i(5)
where H i∑m i
l1h l i h l i
T represents the sample covariance matrix of the surface
S i.By concatenating all the vectors p i T into one vector p p1T p2T p N T T equation(3)can be written as a function of the parameter vector p and we get the following objective function:
F p J p T H p H H100
0H20
00
00H N
(6)
10
Such a function is convex if and only if the matrix H is positive,which is the case.Besides,under the above form,the objective equation contains separate terms for the data and the parameters.The data matrix H can be thus computed off-line before the optimization.
The objective function could be taken as the likelihood of the range data given the parameters(with a negative sign since we want to minimize).The likelihood function has the advantage of accounting for the statistical aspect of the measurements.As a?rst step,we have chosen the least squares func-tion.The integration of the data noise characteristics in the LS function can be done afterwards with no particular dif?culty,leading to the same estimation of the likelihood function in the case of the Gaussian distribution.
4.2.C ONSTRAINT FORMULATION
The different constraints are implemented under a matrix formulation.The matrix notation leads to a compact form and avoids expressions with many variables in particular for the second order derivatives that may be eventually needed in the optmization algorithm.This allows a fast,automatic and easy implementation of the constraints.
Some intrinsic constraints,for instance circularity of quadric surfaces could be imposed implicitly by choosing a suitable form of the surface equa-tion.However,the implementation of the reduced form in the optimization algorithm may cause some complexity.Indeed,because of the nonlinearity of these forms,it has not been possible to get an objective function with separated terms for the data and the parameters.Thus,the data terms could not be computed off-line.This may increase the computational cost dramatically. Examples of how constraints can be implemented are found in section6.
4.3.T HE OPTIMIZATION ALGORITHM
Optimization techniques fall into two broad branches namely Operation Research techniques and the recent evolutionary techniques.
Evolutionary computation techniques[10,11]have been having increas-ing attraction for their potential to solve complex problems.In short they are stochastic optimization methods.They are conveniently presented using the metaphor of natural evolution:they start from a randomly generated set of points or solutions of the search space(population of individuals).Then this set evolves following a process close the natural selection principle.At each stage a new population is generated using simulated genetic operations such as mutation or crossover.The probability of survival of the new solutions de-pends on how well they?t a given evaluation function.The best are kept with high probability and the worst are discarded.This process is repeated until the set of solutions converges to the one best?tting the evaluation function.
11 The main advantages of the evolutionary techniques is that they do not have many mathematical requirements about the optimization problem.They are0-order methods,in the sense that they operate only on the objective function and they can handle linear or nonlinear problems,constrained or unconstrained.
The main drawback of these techniques is that they are highly time consuming.This is due to the fact that to ensure convergence,the number of generated solutions has to be high,and at each iteration all the solutions have to be evaluated.This increases the computation time dramatically.
The second branch of the optimization techniques are the classical op-eration research techniques.They are more mature than the evolutionary techniques.They involve search techniques,numerical analysis and differ-ential tools.Most of these techniques use an iterative scheme.A reasonable initialisation causes signi?cant speedup in convergence.A detailed review and analysis of these optimization techniques could be found in[8,9].
We believe that the evolutionary techniques are suitable mainly to the optimization cases where objective functions and constraints are very com-plex,presenting hard-handled aspects such nonlinearity,non-differentiability, or do have not explicit forms.Indeed the earlier mentioned characteristics of the evolutionary techniques allow them to by-pass these problems.
As our optimization problem does not have these problems,the opera-tional research techniques are more appropriate.This argument is supported by the time-consuming characteristic of the evolutionary techniques,where the average scale of the processing time is on the order of hours.This charac-teristic makes these methods not appropriate for interactive user environments and impractical for a static veri?cation and checking of the results when experiments have to be repeated many times.The other important reason for opting for search techniques is that we can obtain a reasonable initial estimate of the model parameters.This initial solution is the estimation of the model parameters without considering the constraints.This estimation is not far away from the optimal one since it is obtained from the real object prototype.
Theoretically a solution of the problem stated in(2)is given by?nding the set pλ1λ2λk minimizing the following equation:
E p
F p
M
∑
k1
λk C k p
F p p T H p(7)
C k p p T A k p B T k p C k
Under the Khun-Tucker conditions[8](Chapter9),namely that the ob-jective function and the constraint functions are continuously differentiable and the gradients of the constraint functions are linearly independent,the
12
optimal set p λ1λ2
λk minimizing (7)is the solution of the system:
?F
?p
0(8)
In some particular cases it is possible to get a closed form solution for
(8)such as the generalized eigenvalues methods.This depends on the char-acteristics of the constraint functions and whether it is possible to combine them ef?ciently with the objective function.When the constraints are linear (having the form A p B 0)the standard quadratic programming methods could be applied to solve this system.
However the geometric constraints are mainly non-linear.Generally it is not trivial to develop an analytical solution for such problem.In this case an algorithmic numerical approach could be of great help taking into account the increasing capabilities of computing.
Now if we look to the objective function and the constraint functions in (7)we see that they are explicitly de?ned as a function of the paramet-ers,they are smooth,differentiable and they both have a quadratic structure.From (5)we can notice that each submatrix H i of H in (6)is the sum of
cross-product terms h l i h l i T
.Thus H i as well as H are positive de?nite.Con-sequently the objective function is convex.Such functions could be ef?ciently
minimized.Besides it has the important property that its minimum is global.If the constraint functions are squared,thus enforced to be also convex,the optimization problem (7)would be a convex optimization problem for λk 0.For such problem an optimal solution exists,moreover this solution corres-ponds to the solution of the system (8)de?ned by the Khun-Tucker conditions [17](section 27,28).
The problem would be to determine the set p λ1λ2λk minimiz-ing:
E p
F p
M
∑k 1
λk C k
p
2
λk
0(9)
To provide a numerical solution of this problem we have been investigat-ing an approach in the framework of sequential unconstrained minimization.The basic idea is to attach different penalty functions to the objective function F p in such a way that the optimal solutions of successive unconstrained problems approach the optimal solution of the problem (9).Indeed the term
∑M k 1λk C k p
2could be seen as a penalty function controlling the con-straints satisfaction.The scheme then increments the set of λk iteratively,at each step minimize (9)by a standard non-constrained technique,update the solution p ,and repeat the process until the constraints are satis?ed.For equal values of λk ,Fiacco and McCormick [6]have shown that the solutions of (9)converge towards the same solution of the problem (2)when λk tends to in?nity.
13 In more detail the proposed algorithm is:We start with a parameter vec-
tor p0that minimizes the least squares objective function and attempt to?nd
a nearby vector p1that minimizes(9)for small valuesλk.Then we iterat-
ively increase the set ofλk slightly and solve for a new optimal parameter p n1using the previous p n.At each iteration n,the algorithm increases eachλk by a certain amount and a new p n is found such that the optimiz-
ation function is minimized by means of the standard Levenberg-Marquardt
algorithm(see Appendix).The parameter vector p n is then updated to the new estimate p n1which becomes the initial estimate at the next values ofλk.The algorithm stops when the constraints are satis?ed to the desired degree or when the parameter vector remains stable for a certain number of iterations.A simpli?ed version of the algorithm is illustrated in Figure7.a in which a singleλis associated to the constraints.At each iterationλis increased by multiplying it by a factor inversely proportional to the constraint value decrease.
A computational problem associated with this algorithm emerges when
λk become too large.This problem arises in the Hessian matrix of the optim-ization function(9)involved in Levenberg-Marquardt algorithm.This matrix becomes ill-conditioned for high values ofλk.To overcome this problem we have used the technique developed by Broyden et al[3]for updating the parameter vector p at the level of the Levenberg-Marquardt algorithm.
Figure7.Optim:the constraint optimization algorithm.
14
The initialization of the parameter vector is crucial to guarantee the con-vergence of the algorithm to the desired solution.For this reason the initial vector was the one which best?tted the set of data in the absence of con-straints.This vector can be obtained by estimating each surface’s parameter vector separately and then concatenating the vectors into a single one.Nat-urally,the option of minimizing the objective function F p alone has to be avoided since it leads to the trivial null vector solution.On the other hand,the initial valuesλk have to be large enough to avoid the above trivial solution and to give the constraints a certain weight.A convenient value for the initial λk is:
F p0
λ0k
15 Figure8.Structure of the input?le for the constraint language compiler:the upper case words
are the key words of the language
Following the paradigm of Section4.1,each surface is represented by the equation:
h i j T
p i0;i13
h i j x i j y i j z i j1T;p i n i x n i y n i z d i T The object is then represented by the parameter vector: p n1x n1y n1z d1n2x n2y n2z d2n3x n3y n3z d3
16
The objective function is expressed by:
F p J p T H p H10404 04H204 0404H3
where
H i∑
j
h i j h i j T
The surfaces have three orientation constraints re?ecting the three angles 900,900and1200between the three surface normals n1,n2and n3.These constraints are represented by the following equations
n1T n205
n1T n30
n2T n30
from which the constraint functions are deduced:
Angle1p p T A1p0520
Angle2p p T A2p20
Angle3p p T A3p20
where
A1A1i j A1j i12if i1t j5t0t2 A1i j A1j i0otherwise
A2A2i j A2j i12if i1t j9t0t2 A2i j A2j i0otherwise
A3A3i j A3j i12if i5t j9t0t2 A3i j A3j i0otherwise
The surfaces normals are also constrained to be unit.This leads to the following unit constraints:
Unit1p p T U1p120
Unit2p p T U2p120
Unit3p p T U3p120
where U1,U2and U3are diagonal matrices de?ned by
U1U1i i1for i13 U1i i0otherwise
U2U2i i1for i57 U2i i0otherwise
17 Figure9.Input?le of the tetrahedron object.
U3U3i i1for i911 U3i i0otherwise
The expression of the optimization function is then
p T H p
3
∑
l1
λl unit Unit l p
3
∑
l1
λl angle Angle l p
The input?le related to this object is shown in Figure9.
7.Experiments
The experiments were carried out on real parts having planar and quadric sur-faces(cylinder,cone,sphere).The process of extracting the different surfaces of a given part(Fig.10)starts by scanning the part by a3D laser triangulation range sensor.With this device a cloud of3D points representing the shape of the object are obtained.The next step is to segment the points into sets associated to the different surfaces of the object.This is achieved using the rangeseg program[12].To be fully measured,most of the objects have to be scanned at different views.Therefore the measurement data points obtained in each view have to be registered to the same reference frame.This operation is carried out manually by visualising the data points associated to the different views and manipulating the set of points by hand.Since the user relies only on his eye to judge the quality of the registration the data points locations are expected to be additionally corrupted by systematic errors.Actually we have intentionally performed the registration by hand to check the sensitivity of the algorithm with respect to the registration errors.
18
Data capture
Preprocessing
Segmentation and surface fitting
CAD model creation-improvement
Figure10.Steps of the object modelling process
This section will present two experiments carried out on two multi-quadric objects.These experiments check the behaviour and the convergence of the algorithm as well as the impact of constraint satisfaction on the quality of object shape reconstruction.
In order to save some space,the expressions of the different constraints and the way how they were set up will not be developed.The readers can could?nd more details in[21].
7.1.R ECONSTRUCTION EXPERIMENT1
The object(Fig.11)tested in this experiment comprises two lateral planes S1and S2,a back plane S3,a bottom plane S4,a cylindrical surface S5and a conic surface S6.The cylinder surface and the back plane surface contain more than twenty thousands points each.The number of points for each of the other surfaces range from four to nine thousand.The cylindrical patch is less than a half cylinder(40%arc),the conic patch occupies a small area of the whole cone(less than30%)
The surfaces of the object have the following constraints:
1.S1makes an angle of120o with S2(we consider the angle between
normals).
2.S1and S2are perpendicular to S
3.
3.S1and S2make an angle of120o with S
4.
4.S3is perpendicular to S4.
5.The axis of the cylindrical patch S5is parallel to S3’s normal.
6.The axis of the cone patch S6is parallel to S4’s normal.
7.The cylindrical patch is circular.
8.The cone patch is circular.
19
(a)
(b)
(c)(d)
lateral plane
S1
cylinder patch
S2
cone patch
S5
S6
back plane
S3
bottom plane
S4
lateral plane
Figure11.four views of the multi-quadric object
Constraints5and6are imposed by associating the normals to S3and S4 respectively to the orientation vectors of the cylinder axis and the cone axis and thus could be combined with the angle constraints(see[21]for explicit development).
The complete optimisation function is then given by the expression:
E p p T H pλ1C unit pλ2C ang pλ3C circ cyl pλ4C circ cone p Since the surfaces cannot be recovered from a single view,four views(Fig.11) have been registered by hand.100estimations were carried out for statistical comparison.At each trial50%of the surface’s points are selected randomly. The results shown in this section are the average of these estimations.The results regarding the algorithm convergence are shown in Figure12.The be-haviour of the different constraints during the optimization have been mapped as a function of the associatedλi as well as the least squares residual and the sum of the constraint functions.The?gures show a nearly linear logar-ithmic decrease of the constraints.It is also noticed that at the end of the optimization all the constraints are highly satis?ed.The least squares error converges to a stable value and the constraint function vanishes at the end of the optimization.Thus,the?nal part shape satis?es the shape constraints at a slight increase in the least squares?tting error.The?gures also show that it is possible to continue the optimization further until a higher tolerance is
20
reached,however this is limited practically by the numerical accuracy of the machine.
(e)(f)
Figure12.a,b,c,d:Decrease of the different constraint errors as function of the relatedλ.e,f :Variation of the least squares error and the total constraint error.
Are you blind when you're born? Can you see in the dark? 你出生时是目盲的吗？你能在黑暗处看到事物吗？ Dare you look at a king? Would you sit on his throne? 你敢目室国王吗？你想坐在王位上吗？ Can you say of your bite that it's worse than your bark? 你能说你的咬力不如你的叫声吗？ Are you cock of the walk when you're walking alone? 当你独自行走时你很自信自傲吗？ Because jellicles are and jellicles do 因为杰利克是杰利克能 Jellicles do and jellicles would 杰利克能杰利克会 Jellicles would and jellicles can 杰利克会杰利克能 Jellicles can and jellicles do 杰利克能杰利克做 When you fall on your head, do you land on your feet? 当你头朝下落下时，你能用脚着地吗？ Are you tense when you sense there's a storm in the air? 当你感到风暴来临时，你会很紧张吗？ Can you find your way blind when you're lost in the street? 当你迷路时，你能本能的找到正确的方向吗？ Do you know how to go to the heaviside layer? 你知道如何升向九重天吗？ Because jellicles can and jellicles do 因为杰利克能杰利克做 Jellicles do and jellicles can 杰利克做杰利克能 Jellicles can and jellicles do 杰利克能杰利克做 Jellicles do and jellicles can 杰利克做杰利克能 Jellicles can and jellicles do 杰利克能杰利克做 Can you ride on a broomstick to places far distant? 你能骑着扫把去很远的地方吗？ Familiar with candle, with book, and with bell? 你喜爱玩耍蜡烛，书籍或是铃铛吗？ Were you Whittington's friend? The Pied Piper's assistant? 你是惠廷顿的朋友？或是吹笛手的助理吗？ Have you been an alumnus of heaven and hell? 你能自由的通往天堂和地狱吗？ Are you mean like a minx? Are you lean like a lynx? 你是一个爱出风头的姑娘吗？你瘦的像一只山猫吗？ Are you keen to be seen when you're smelling a rat? 当你闻到一只老鼠你会努力的寻找吗？ Were you there when the pharaoh commissioned the Sphinx? 当法老委派做狮身人面像时你在场吗？ If you were, and you are, you're a jellicle cat 如果你在并且你是，你是一只杰利克猫． Jellicle songs for jellicle cats 杰利克歌咏杰利克猫 Jellicle songs for jellicle cats 杰利克歌咏杰利克猫 Jellicle songs for jellicle cats 杰利克歌咏杰利克猫
我的歌声里 没有一点点防备 也没有一丝顾虑 你就这样出现 在我的世界里带给我惊喜情不自己可是你偏又这样 在我不知不觉中悄悄地消失 从我的世界里没有音讯 剩下的只是回忆 你存在我深深的脑海里 我的梦里我的心里我的歌声里 你存在我深深的脑海里 我的梦里我的心里我的歌声里 还记得我们曾经 肩并肩一起走过那段繁花巷口 尽管你我是陌生人是过路人 但彼此还是感觉到了 对方的一个眼神一个心跳 一种意想不到的快乐 好像是一场梦境命中注定 你存在我深深的脑海里 我的梦里我的心里我的歌声里 你存在我深深的脑海里 我的梦里我的心里我的歌声里 世界之大为何我们相遇 难道是缘分难道是天意 你存在我深深的脑海里 我的梦里我的心里我的歌声里 你存在我深深的脑海里 我的梦里我的心里我的歌声里 你存在我深深的脑海里 我的梦里我的心里我的歌声里 如水 期待过我们似细水 可惜蒸发出眼泪 明白你最近有些暂时伴侣偷一刻午睡彷佛专一使你极空虚 怀疑被你抱着我念着谁 无论你再好亦舍得失去 难过亦过难道我 嫌损失未够多
早放手可减轻痛楚 不等泡沫给吹破 不想去知谁填补我 无悔在我还是我 任你多么差错 无谓去追问为何 深知告别损失非我 让情人离别 似水清洗我 原谅你对着我说谎 出于好意的作状 明白你最近已经避谈近况 早不敢寄望 心中早把相爱如观光 情如瀑布泻下也未惊慌 心境已随着那水花得到释放 难过亦过难道我 嫌损失未够多 早放手可减轻痛楚 不等泡沫给吹破 不想去知谁填补我 无悔在我还是我 任你多么差错 无谓去追问为何 深知告别损失非我 让情人离别 似水清洗我 难过亦过难道我 嫌损失未够多 早放手可减轻痛楚 不等泡沫给吹破 不想去知谁填补我 无悔在我还是我 任你多么差错 无谓去追问为何 深知告别损失非我 让情人离别 似水清洗我 心中有涟漪吹过又回到最初平静去做我
歌曲背景 "Hallelujah" is a song written by Canadian singer Leonard Cohen, originally released on his album Various Positions (1984). Achieving little initial success, the song found greater popular acclaim through a recording by John Cale, which inspired a recording by Jeff Buckley. It has been viewed as a "baseline" for secular hymns. "Hallelujah"为加拿大著名游吟诗人、民谣歌手Leonard Cohen在1985年创作的歌曲，收录在其专辑"Various Positions"中。其歌词充满诗意，内涵丰富，曲调缓慢忧伤，加上Leonard沧桑嗓音的低吟浅唱，演绎出了一种清淡而悠长的回味。 Hallelujah的版本很多，其中最有影响力的还是美国著名创作型歌手Jeff Buckley的翻唱版本，被收录在其1994年的专辑"Grace"中。Jeff被U2的Bono形容为“噪海中的纯净一滴”，他的声音明丽甜蜜又性感飘渺，诠释起悲伤和记忆来却更令人印象深刻。 中国歌手邓紫棋在现场演唱会上翻唱了本曲。 英文歌词 Now I've heard there was a secret chord 我曾听闻一曲传奇中的旋律 That David played, and it pleased the Lord 是大卫弹奏来取悦上帝的赞颂 But You don't really care for music, do You? 但祢真正喜悦的(是人的作为，而)不是音乐，对吧？ Well it goes like this 旋律是这样的 The fourth, the fifth F和弦，G和弦(复杂的心情无可言喻) The minor fall, the major lift 大小调起承转合(指戴维泣不成声的祷告颂唱，至五音不全) The baffled king composing Hallelujah
1A 1.Are You Sleeping? Are you sleeping? Are you sleeping? Brother John, Brother John? Morning bells are ringing, Morning bells are ringing. Ding, ding, dong! Ding, ding, dong! 2.Ten Little Indians One little, two little, three little Indians,
Four little, five little, six little Indians, Seven little, Eight little, Nine little Indians, Ten little Indian boys. One little, two little, three little Indians, Four little, five little, six little Indians, Seven little, Eight little, Nine little Indians, Ten little Indian girls. 3.Happy Birthday to You Happy birthday to you. Happy birthday to you. Happy birthday, dear friend. Happy birthday to you.
4.Hot Potato One potato, two potatoes, Three potatoes, four, Five potato, six potatoes, Seven potatoes, more. 5.Hot Cross Buns Hot cross buns! Hot cross buns! One a penny, two a penny. Hot cross buns! If you have no daughters, Give them to you sons!
歌名：As Long As You Love Me 歌手：Justin Bieber 所属专辑：Believe Acoustic 作曲 : Persson Svensson 作词 : Persson Svensson As long as you love me yeah 只要你爱我就好 I'm under pressure, seven billion people in the world trying to fit in 我们在压力下跟着全世界70亿人适应这个社会 Keep it together, smile on your face even though your heart is frowning 紧紧相依，你心有困懑却面带笑容 But hey now, you know girl, we both know it's a cruel world 但是现在，宝贝你知道，我们都知道世界多么残酷 But I will take my chances 但我愿意（搏一搏）抓住我的机会 As long as you love me, we could be starving, 只要你爱我，我们可以挨饿（饥肠辘辘） We could be homeless, we could be broke 可以流离失所，也可以支离破碎 As long as you love me I'll be your platinum, I'll be your silver, i'll be your gold 只要你爱我，我是你的铂金，我是你的银，我是你的财富（我会不离不弃，无坚不摧，所向披靡）
1、《Mister Sun》 Oh Mister Sun, Sun, Mister Golden Sun, Please shine down on me. Oh Mister Sun, Sun,Mister Golden Sun, Hiding behind a tree… These little children ， Are asking you To please come out So we can play with you. Oh Mister Sun, Sun,Mister Golden Sun, Please shine down on me! Oh Mister Sun, Sun, Mister Golden Sun, Please shine down on me. Oh Mister Sun, Sun, Mister Golden Sun,Hiding behind a tree…These little children Are asking youTo please come out So we can play with you. Oh Mister Sun, Sun,Mister Golden Sun, Please shine down on,Please shine down on, Please shine down on me! 2、Mango Walk My brother did a tell me That you go mango walk You go mango walk(2×) My brother did a tell me That you go mango walk And pick all the numbe r ?leven 3、Mosquito Mosquito one, Mosquito two. Mosquito jump in the old man ,shoe Zzzzzzzz(clap the mosquito ) 4、Artist：harrybelafonte Songs Title：coconut woman （Coconuts, coconuts） （Coconuts, coconuts） Coconut woman is calling out And everyday you can hear her shout Coconut woman is calling out And everyday you can hear her shout Get your coconut water （Four for five） Man, it's good for your daughter （Four for five） Coco got a lotta iron （Four for five） Make you strong like a lion （Four for five） A lady tell me the other day No one can take her sweet man away I ask her what was the mystery She say coconut water and rice curry You can cook it in a pot （Four for five） You can serve it very hot （Four for five） Coco got a lotta iron （Four for five） Make you strong like a lion （Coconuts, coconuts） Coconut woman says you'll agree Coconut make very nice candy The thing that's best If you're feeling glum Is coconut water with a little rum It could make you very tipsy （Four for five） Make you feel like a gypsy （Four for five） Coco got a lotta iron （Four for five） Make you strong like a lion （Four for five） Ah, play that thing Coconut woman is calling out And everyday you can hear her shout Coconut woman is calling out And everyday you can hear her shout Get your coconut water （Four for five） Man, it's good for your daughter （Four for five） Get your coconut candy （Four for five） Make you feel very dandy （Four for five） Coco, coco, coco... Coconut, coconut Coconut, coconut... -
听的圣诞节英语歌曲 下面是一些圣诞节英文歌曲推荐！ 1、Everybody Knows I Love U 圣诞表白 2、Hallelujah-Alexandra Burke圣诞单曲销售冠军 3、圣诞必听英文歌曲John Lennon(Happy Christmas) 4、豌豆公主Kylie Minogue圣诞情调Santa Baby 5、小天后Taylor Swift演绎缤纷圣诞Last Christmas 6、NIKE最新圣诞广告歌曲完整版 7、Lady GaGa 欢乐圣诞单曲Christmas Tree 8、最畅销圣诞歌Mariah Carey(All I Want For Christmas Is You) 9、超人气童星Justin Bieber在奥巴马总统面前献唱圣诞歌曲 10、可爱童声贺圣诞When Christmas comes to town 欢乐合唱团相约快乐圣诞 Glee Cast - Last Christmas 美国福克斯电视台于2009年推出的热门青春音乐剧《欢乐合唱团》Glee盛邀剧中的男女主演们，包括Lea Michele, Amber Riley, Cory Monteith, Kevin McHale, Jenna Ushkowitz, Chris Colfer, Dianna Agron, Mark Salling等一席欢乐合唱团的团员们齐齐出动，为我们带来了这首欢快的经典翻唱"Last Christmas"。 绯闻女孩Queen B喊你回家过圣诞 Leighton Meester - Christmas "Christmas (Baby Please Come Home)"是Darlene Love于1963年发表的经典圣诞歌曲，后被乐队U2、乐坛天后Mariah Carey翻唱。
I've heard there was a secret chord That David played, and it pleased the Lord But you don't really care for music, do you? It goes like this The fourth, the fifth The minor fall, the major lift The baffled king composing Hallelujah Hallelujah, Hallelujah Hallelujah, Hallelujah Your faith was strong but you needed proof You saw her bathing on the roof Her beauty in the moonlight overthrew you She tied you to a kitchen chair She broke your throne, and she cut your hair And from your lips she drew the Hallelujah Hallelujah, Hallelujah Hallelujah, Hallelujah Baby I have been here before I know this room, I've walked this floor I used to live alone before I knew you. I've seen your flag on the marble arch Love is not a victory march It's a cold and it's a broken Hallelujah Hallelujah, Hallelujah Hallelujah, Hallelujah Maybe there’s a God above But all I’ve ever learned from love Was how to shoot at someone who outdrew you It’s not a cry you can hear at night It’s not somebody who has seen the light It’s a cold and it’s a broken Hallelujah Hallelujah, Hallelujah Hallelujah, Hallelujah I did my best, it wasn't much I couldn't feel, so I tried to touch I've told the truth, I didn't come to fool you And even though it all went wrong I'll stand before the Lord of Song With nothing on my tongue but Hallelujah
Miley Cyrus - The Climb / 麦莉·赛勒斯- 攀登 I can almost see it / 眼前依稀浮现 That dream I'm dreamin' but / 萦绕心头的那个梦境There's a voice inside my head saying / 脑海里却响起一个声音 you'll never reach it / 你永远也不会到达彼岸Every step I'm taking / 我迈出的每一步 Every move I make feels /我做过的每件事 Lost with no direction / 无不使我迷失方向 My faith is shakin / 开始动摇的，是我的信念 But I, I gotta keep tryin. / 可我，我还是要继续求索Gotta keep my head held high / 还是要挺胸抬头、阔步前行 There's always gonna be another mountain / 总会有下一座山峦 I'm always gonna wanna make it move / 在等我去将它移开 Always gonna be an uphill battle / 总会有下一个山坡 Sometimes I'm gonna have to lose / 很可能令我无法越过 Ain't about how fast I get there / 不在于我要用多久才能抵达峰顶 Ain't about what's waitin on the other side / 不在于山那边倒底是怎样的风景 It's the climb / 这就是攀登 The struggles I'm facing / 我面对的每次搏击 The chances I'm taking / 我抓住的每次机遇Sometimes might knock me down but / 有时会令我一败涂地 No I'm not breaking / 却决不会磨去我的意志 I may not know it / 或许我不懂其中的意义 But these are the moments that / 但这些时刻却会成为 I'm gonna remember most, yeah / 我一辈子可以珍藏的回忆，啊 Just gotta keep going / 只管继续前进 And I, I gotta be strong / 我要，我要变得坚强 Just keep pushing on 'cause / 只须奋力前行，因为 There's always gonna be another mountain / 总会有下一座山峦I'm always gonna wanna make it move / 在等我去将它移开 Always gonna be an uphill battle / 总会有下一个山坡 Sometimes I'm gonna have to lose / 很可能令我无法越过 Ain't about how fast I get there / 不在于我要用多久才能抵达顶峰 Ain't about what's waitin on the other side / 不在于山那边倒底是怎样的风景 It's the climb / 这就是攀登 Yeah-yeah / 啊- There's always gonna be another mountain / 总会有下一座山峦 I'm always gonna wanna make it move / 在等我去将它移开 Always gonna be an uphill battle / 总会有下一个山坡 Sometimes you're gonna have to lose / 很可能令我无法越过 Ain't about how fast I get there / 不在于我要用多久才能抵达顶峰 Ain't about what's waitin on the other side / 不在于山那边倒底是怎样的风景 It's the climb / 这就是攀登 Yeah-yeah-yeah / 啊- - Keep on moving / 继续前行 Keep climbing / 继续攀登 Keep the faith / 坚守信念 Baby / 宝贝 It's all about / 这一切就是 It's all about the climb / 这一切就是攀登
Hallelujah Well I heard there was a secret chord 我听说有个神秘的和弦 That David played, and it pleased the Lord 大卫弹奏以取悦主 But you don't really care for music, do ya? 可你并不关心音乐，不是么 Well it goes like this 它这样奏起 The fourth, the fifth 四度，五度(2)
The minor fall and the major lift 小调降，大调升The baffled king composing Hallelujah 徒然哀求的君王谱下哈利路亚(3) Hallelujah Hallelujah 哈利路亚哈利路亚 Hallelujah Hallelujah 哈利路亚哈利路亚Well Your faith was strong but you needed proof 你信心坚定但需受考验 You saw her bathing on the roof 你在屋顶见她沐浴 Her beauty and the moonlight overthrew you 她在月光下的美丽将你击溃(4) She tied you to her kitchen chair 她将你捆在厨房的椅上 And she broke your throne and she cut your hair 她毁了你的王位剪下你的头发(5) And from your lips she drew the Hallelujah 从你的唇中她吸吮哈利路亚 Hallelujah Hallelujah 哈利路亚哈利路亚 Hallelujah Hallelujah 哈利路亚哈利路亚
Rufus Wainwright - Hallelujah 歌词翻译： I heard there was a secret chord that David played and it pleased the Lord But you don’t really care for music, do you? Well it goes like this : The fourth, the fifth, the minor fall and the major lift The baffled king composing Hallelujah 我听见了那神秘悠扬的旋律 那是以色列王（David）为取悦上帝而奏 但也许你并不在意旋律本身，不是吗？ 音乐却是这样起来的 第4，第5，小调落下，大调升起 饱受煎熬的国王写下了赞美之歌－哈利路亚 Hallelujah Hallelujah Hallelujah Hallelujah… 哈利路亚，哈利路亚，哈利路亚““` Your faith was strong but you needed proof You saw her bathing on the roof Her beauty and the moonlight overthrew you And she tied you to her kitchen chair She broke your throne and she cut your hair But from your lips she drew the Hallelujah 你信念坚定却也要受到考验 你在屋顶上看到她在那里沐浴 她的美貌在月光下就已经把你征服 她会把你骗到坐上厨房里的椅子上 推翻你的宝座，并剪下你的发丝 为了听到你的唇边的赞美之歌－哈利路亚 Hallelujah Hallelujah Hallelujah Hallelujah… 哈利路亚，哈利路亚，哈利路亚“““` Baby I’ve been here before I’ve seen this room and I’ve walked this floor I used to live alone before I knew ya And I’ve seen your flag on the marble arch
Turn off the Lights 70首唯美欧美歌推荐 Ruolin_run 很久没有写文了，近来无事，方生一念，将库存里的歌整一整，分了类，挑出这70首唯美曲。先说明，这70首曲风绝对是极唯美的，绝大多数属于英文民谣或蓝调，有部分是小众派，大多数以钢琴、吉他伴奏，先后顺序无优劣之分，请关上灯，戴上耳机，不要受打扰，自己一个人听。 噢…再说一句，你看我打了这么多，路过的，请留贴 1.30 Minutes-T.A.T.U 曾经风靡一时的俄罗斯少女二重唱组合为我们带来的纯唱…你一定会爱上的…好像夜空中划过的流星..歌曲以音乐的笑声开场，带出旋律，伴奏不显眼，最后全曲又不知不觉消失… 2.A Perfecr Indian-Sinead O’Connor 全曲以钢琴伴奏开场，你听懂了歌手的吟语吗…她唱出了一种独特的孤独…一个年轻就剃光头的女子必是孤独的… 3.A Place Nearby-Lene Marlin 挪威小才女Lene Marlin的代表作，轻轻哼唱，没有震撼感，就像你的邻家小女孩的哼唱一样，但绝对能攫住你的心 4.Black Star-Avril Lavigne 歌手绝对是众所周知的了，但歌曲却不怎么著名…这是Avril歌中最抒情的一首之一，以近似八音盒的乐器开场，清脆，悠扬… 5.Bluebird-Christina Perri 我居然把我最喜欢的歌手排到第五…全曲基本以吉他为主旋律伴奏，曲调轻快悠扬，富有生机…歌手是民谣歌手，目前不著名，但是底子很厚，但民谣歌手的身份，让她不容易红起来… 6.Breathe Me-Sia 歌手声音稍稍沙哑，呓语恰与钢琴伴奏的欢快明亮形成了对比，别有独特的滋味… 7.Butterfly Fly Away-Miley Cyrus 这个肯定很多人听过啦…类似于吉他的自弹自唱，高潮部分的父女双声部合唱更是感人，是一个少女对未来的憧憬… 8.By Our Love-Christy Nockels 这其实是宗教歌曲，但如果你没查歌词…是看不出来的，曲风庄严唯美，感人至深…舒缓的钢琴旋律配合着歌手醇厚的嗓音，缓缓淌出…听一遍，就会被迷住… 9. Caresse Sur L'Ocean-Kelly Sweet 这首《放牛班的春天》的主题曲，经过甜心Kelly的演绎，褪去了少年的青涩，变得更为纯美，同样空灵澄澈… 10.Childhood-Michael Jackson 歌手太著名，致敬…唱出了一个唯美的理想儿童世界，MV中的Michael却一人独自流出感伤之情，是对自己的童年的感伤么…曲风唯美感伤，Michael声音中的空灵发挥到了极致，全曲有电影艺术的感觉… 11．Come Away With Me-Norah Jones 这个歌手也太著名…爵士才女，这是她的代表作，很能体现她像葡萄酒冰激凌一样醇厚的嗓音… 12.Corner of Your Heart-Ingrid Michaelson 与前面的A Perfect Heart曲风相似，都有种不经意间的孤独之感…女声空旷灵澈，属于多听几遍才会爱上的类型… 13.Craigie Hill-Cara Dillon
《Hallelujah》背后的故事： Hallelujah，哈利路亚，希伯来语，意为“赞美上帝”。邓紫棋演唱的这首《Hallelujah》，是加拿大著名诗人/作家/歌手Leonard Cohen，在1984年的专辑《Various Positions》中那首最为著名的同名单曲。 《Hallelujah》歌词几乎是处处依托圣经故事的。比如在第一段歌词中提到的David。David，大卫，牧羊人出身，但容貌俊美，英勇善战，能赋能歌，战胜巨人哥利亚，被上帝选为以色列的国王。在大卫当国王的时候，一日傍晚，在阳台闲逛时偶然发现一貌美女子正在沐浴（《Hallelujah》歌词中的You saw her bathing on the roof / Her beauty and the moonlight overthrew you），娇媚性感的曲线加上柔和迷人的月光将大卫彻底吸引住。之后，大卫便派人四处打听此女子，想要知道她的芳名。原来，这个迷人的女子名叫Bethsheba （拔示巴），是大卫的一位部将Uriah（乌利亚）的妻子。大卫忍受不住占有拔示巴的欲望，终于让人将拔示巴接入宫中，与其同房。不久，拔示巴怀上了大卫的孩子，大卫恐慌万分，急忙召回远在前线的乌利亚，借口让其休整一下，意图让乌利亚回到家中与妻子同房以掩盖拔示巴肚中孩子的亲缘归属。 但是，乌利亚是一位忠于职守为国尽忠的勇士，他对大卫说：“国家处于危机，我怎有心思回家与妻子相欢，溺于安享。”乌利亚要求大卫将自己重新派回到战场上。大卫此时便顺水推舟，并写成密信一封，让乌利亚携信到大将军约亚那里；信中大卫指示约亚将乌利亚派往最为危险的战场。于是，乌利亚就在大卫的密谋和意愿下战死沙场，而大卫则将拔示巴迎进自己的后宫，名正言顺的占有了拔示巴。 然而，占有拔示巴的大卫并不快乐，他白天对着众人强颜欢笑，夜晚就成为自己罪过的奴隶，受尽折磨。大卫认识到自己的罪行，写下诗篇五十一篇忏悔书，歌与上帝，祈求宽恕（Now I've heard there was a secret chord / That David played, and it pleased the Lord / But you don't really care for music, do you?） “She tied you T o a kitchen chair / She broke your throne, and she cut your hair” 这个场景则是Samson（参孙）和Delilah（达利拉）的故事。以色列在被非力士人统治的时候，族中有个大力士，名叫参孙，上帝赐予他空手撕裂雄狮的神力，这使得非力士统治者颇为惧怕，但是，参孙有个弱点：上帝与其约定——不得剪发，如是，则神力尽失。参孙爱上了一位非力士族的姑娘，名叫达利拉。非力士统治者赐予达利拉许多钱财，命令她与参孙相好并套出他为何力大无穷的秘密。参孙在达利拉的三次诱惑下以编造的故事将其蒙混，但是最后参孙还是将自己的秘密告诉的达利拉。一日，达利拉趁安抚参孙熟睡于自己的膝上时，让人将参孙的头发剪掉了。参孙醒来，发现自己的头发被剪，神力也一并消失了。可怜的参孙在爱情的迷惑下，违背了与神定下的誓言，从一个英勇的大力士变成了一个人人欺辱的瞎子。这首歌的歌词里，也描写了作为罪人在罪恶中的迷茫、迷惘和空虚失落；被罪恶所辖制的痛苦；而要要从罪中结果，脱离罪的辖制，惟有借着主耶稣宝贵血的恩典才可以做到；也许这正是邓紫棋在那见证中所经历的过程；但你重回主耶稣的怀抱中时，一切的不安、迷惘、空虚都离你而去了，从此你得到满足。 那时，你就会情不自禁的开口赞美主，如作者Leonard Cohen所说，“当一个人展望世界回首生命时，唯一可言的词语便是Hallelujah”。 创作信息 "Hallelujah"为加拿大著名游吟诗人、民谣歌手Leonard Cohen在1985年创作的歌曲，收录在其专辑"Various Positions"中。其歌词充满诗意，内涵丰富，曲调缓慢忧伤，加上Leonard 沧桑嗓音的低吟浅唱，演绎出了一种清淡而悠长的回味。 Jeff Buckley翻唱版本
hallelujah I heard there was a secret chord 我听见了那神秘悠扬的旋律 that David played and it pleased the Lord 那是以色列王（David）为取悦上帝而奏 But you don't really care for music, do you? 但也许你并不在意旋律本身，不是吗？ Well it goes like this : 音乐却是这样起来的 The fourth, the fifth, the minor fall and the major lift 第4，第5，小调落下，大调升起 The baffled king composing Hallelujah 饱受煎熬的国王写下了赞美之歌－哈利路亚 Hallelujah HallelujahHallelujahHallelujah... 哈利路亚，哈利路亚，哈利路亚````` Your faith was strong but you needed proof 你信念坚定却也要受到考验 You saw her bathing on the roof 你在屋顶上看到她在那里沐浴 Her beauty and the moonlight overthrew you 她的美貌在月光下就已经把你征服 And she tied you to her kitchen chair 她会把你骗到坐上厨房里的椅子上 She broke your throne and she cut your hair 推翻你的宝座，并剪下你的发丝 But from your lips she drew the Hallelujah 为了听到你的唇边的赞美之歌－哈利路亚 Hallelujah HallelujahHallelujahHallelujah... 哈利路亚，哈利路亚，哈利路亚``` ````
ハレルヤ（哈利路亚） 专辑：Promised Land 歌手：RURUTIA （ルルティア） 作词：RURUTIA （ルルティア） 作曲：RURUTIA （ルルティア） 翻译：风之阡陌 远くこだまするは獣たち/ 远处传来的声音是野兽们 深い夜に罪を笑い语る/ 在深夜里狞笑着谈论罪行 血涂られた正义/ 血染的正义 汚れた身体を/ 污秽的身体 锖びた雨が磨く/ 被生锈的雨水刷洗 重ねた过ちた饰られた街に/ 装饰着重复罪过的街道上 死の灰が降る/ 死之灰烬降下 甘く忍びよるは魔物たち/ 天真潜近的妖魔们 穴の开いた胸に笑いかける/ 朝着打开的胸口冷笑 堕落する天使/ 堕落的天使 圣なる翼と替えた/ 与圣洁羽翼交换的 禁断の杯/ 禁忌之杯 涡巻く欲望は加速した/ 被加速卷起漩涡的欲望 街は沈んでいく/ 渐渐地沉没在街道上 OH ハレルヤ全て洗い流して/ OH 哈利路亚将一切冲洗吧
汚れた身体を/ 污秽的身体 锖びた雨が磨く/ 被生锈的雨水刷洗 重ねた过ちた饰られた街に/ 装饰着重复罪过的街道上死の灰が降る/ 死之灰烬降下 圣なる翼と替えた/ 与圣洁羽翼交换的 禁断の杯/ 禁忌之杯 涡巻く欲望は加速して/ 被加速卷起漩涡的欲望 街は沈んでいく/ 渐渐地沉没在街道上 汚れた身体を/ 污秽的身体 锖びた雨が磨く/ 被生锈的雨水刷洗 重ねた过ちた饰られた街に/ 装饰着重复罪过的街道上死の灰が降る/ 死之灰烬降下 to o ku ko da ma su ru wa ke mo no ta chi fu ka ku yo ru ni tsu mi wo wa ra i ka ta ru chi nu ra re ta se i gi yo go re ta ka ra da wo sa bi ta a me ga mi ga ku ka sa ne ta a ya ma chi de ka za ra re ta ma chi ni shi no ha i ga fu ru a ma ku shi no bi yo ru wa ma mo no ta chi a na no a i ta mu ne ni wa ra i ka ke ru
最经典的五首英文歌曲 不仅旋律优美动听，而且歌词可读性高。1．only love Two a m and the rain is falling Here we are at the crossroads once again You're telling me you're so confused You can't make up your mind Is this meant to be You're asking me But only love can say Try again or walk away But I believe for you and me The sun will shine one day So I just play my part Pray you'll have a change of heart But I can make you see it through That's something only love can do In your arms as the dawn is breaking Face to face and a thousand miles apart I've tried my best to make you see There's hope beyond the pain If we give enough If we learn to trust
But only love can say Try again or walk away But I believe for you and me The sun will shine one day So I just play my part Pray you'll have a change of heart But I can make you see it through That's something only love can do I know if I could find the words To touch you deep inside You'll give my dreams just one more chance To let this be our last goodbye But only love can say Try again or walk away But I believe for you and me The sun will shine one day So I just play my part Pray you'll have a change of heart But I can make you see it through That's something only love can do That's something only love can do