CN102930586A - Interactive geometry deformation method based on linear rotation invariant differential coordinates - Google Patents

Abstract

The invention discloses an interactive geometry deformation method based on linear rotation invariant differential coordinates and belongs to the field of computer virtual reality technology. The interactive geometry deformation method comprises the following steps of: firstly decomposing a differential structure of a grid model to obtain a rigid body motion invariant in grids, namely a discrete basic factor, and then establishing a local frame of a model midpoint for a normal vector at any point in the grids and the relationship between the point and surrounding points for solving subsequent deformation results. The interactive geometry deformation method establishes the linear rotation invariant differential coordinates, so that wide-angle transformation such as translation and rotation for a curved surface can be interactively restrained by a user in real time so as to achieve the purposes of adjusting and modifying products by the user in real time through deformation operation of a three-dimensional grid model in product design and animated character creation.

Description

A kind of alternately geometry deformation method based on the constant differential coordinate of Linear Rotation

Technical field

The invention belongs to the computer virtual reality technology field, particularly a kind of alternately geometry deformation method based on the constant differential coordinate of Linear Rotation is applied in the Computer Animated Graph.

Background technology

Along with computer science in the develop rapidly aspect the hardware and software, computer graphics is all being played the part of important role in the every field of modern society.Existing simple objects is out of shape to obtain complex object, and realizes alternatively that by deformation technology animation effect becomes the focus and emphasis of nearest graphics area research especially.The hot topic in geometric manipulations field studies a question how to keep the interactive gridding deformation technology of model detail to become in recent years.

Existing distortion of the mesh work roughly is divided into the distortion of Free Transform, multiresolution and is out of shape 3 classes based on differential domain from geometric angle.The Free Transform technology is embedded into object in the container, reaches distortion and editor's purpose by process container.Although the Free Transform technology has the advantage of being convenient to user interactions and highly effective, be difficult to keep the geometric detail of model.Multi-level technique is smooth base net lattice and high frequency details with former grid decomposition, then the high frequency details is superimposed upon on the base net lattice after the distortion and obtains distorted pattern.Although the method can keep the local geometric details of model, when the overlapped high-frequency details, do not consider base net lattice relation with it, the phenomenons such as details distortion can appear when moderate finite deformation.The differential domain distortion of the mesh technology that comes into vogue is in recent years compared with the above two has the performance that keeps details, and increasing researchist both at home and abroad just constantly drops in the research of the method.

Deformation technology based on the differential coordinate represents geometric detail with local differential coordinate, distortion is considered as an energy-optimised problem, satisfying the differential coordinate that keeps as much as possible these reaction geometric detail under the condition of user restraint, the differential coordinate that comes match to change by finding the solution constant coefficient linearity system of equations or nonlinear optimization.Because of its simple, healthy and strong performance advantage, become the main flow algorithm of in recent years distortion of the mesh based on the distortion of the mesh of differential.

But owing to the differential coordinate that is defined in global coordinate system is not invariable rotary, only keep the constant distortion of the mesh meeting of differential coordinate to produce the distortion of distortion.Traditional work is around the differential coordinate that how better to approach warp mesh, comparatively suitable deformation matrix of implicitly match of angle of generally optimizing from a least square, this method does not tackle the problem at its root, when being out of shape, big angle rotary can produce serious distortion, can not keep the vision confidence level of three-dimensional grid model, so that three-dimensional grid model editing operation effect is affected.

Summary of the invention

Technology of the present invention is dealt with problems: in order to overcome the serious distortion that produces when big angle rotary is out of shape, reach preferably deformation effect, a kind of alternately geometry deformation method based on the constant differential coordinate of Linear Rotation has been proposed, the method is set up invariable rotary differential coordinate, simultaneously can adopt interactive mode to provide constraint by the user in real time, make the user when carrying out product design, cartoon role creation, can by three-dimensional grid model is carried out deformation operation, reach the purpose that product is adjusted in real time and revised.

Technical solution of the present invention: a kind of alternately geometry deformation method based on the constant differential coordinate of Linear Rotation, its characteristics are that step is as follows:

(1) calculates the discrete basic parameter of original mesh model curved surface

Differential structrue to original mesh model curved surface is decomposed, and extracts the rigid motion invariant that the coordinate system choosing and put with the parameter of putting all has nothing to do, and is called discrete basic parameter; Described discrete basic parameter will be as local frame and the point equation of setting up the grid model mid point;

(2) the local frame of foundation point

For any one point in the original mesh model, the principal direction according to normal vector and this point of this point constructs the right-handed Cartesian coordinate system that initial point places this point, is called local frame a little; Local frame phasor difference in the computing grid model curved surface between any two consecutive point, the local frame that obtains between the consecutive point is poor;

(3) determine and find the solution the Discrete Surfaces equation

Poor according to the local frame between the arbitrary neighborhood point in the grid model curved surface, determine the Discrete Surfaces equation, the solving result of Discrete Surfaces equation is called new local frame a little for the local frame of the grid model mid point after being out of shape;

(4) set up point equation

For any point in the grid model, with the discrete basic parameter in the original mesh model of obtaining in the step (1), regard the coordinate of vector in the new local frame of this point that this consists of to these consecutive point as; Relation between the consecutive point consists of a system of equations, and the unknown number of system of equations is the coordinate of point after the distortion, and this system of equations is called point equation;

(5) calculated deformation effect

Customer interaction information as binding occurrence, is generated point equation group right-hand member vector and carries out C＋＋ language; Result of calculation is the position of the rear grid model mid point of distortion; Adopt interactive mode to provide constraint by the user in real time, make the user when carrying out product design, cartoon role creation, by three-dimensional grid model is carried out deformation operation, reach the purpose that product is adjusted in real time and revised.

Being implemented as follows of described step (1):

(1) for original mesh model curved surface, the section of any point is determined by the unit normal vector of this point in the described curved surface, the unit normal vector of this point is the linear combination of two parameter local derviation vectors, the parameter of the section of any point and this point is chosen irrelevantly in this grid model curved surface, and the section of point can be used as calculates the rigid motion invariant;

(2) with near the original mesh model curved surface Taylor expansion arbitrfary point, the Taylor form most end includes the high-order infinitesimal of parameter secondary increment, do not considered, near the point this point on the replacement curved surface that can be similar to of the point on this grid model curved surface then, the section of this point is a section of original mesh model; Determine a quadric surface by a section; Quadric surface is called the secondary tangent plane in the projection of the method direction of original mesh model, and the secondary tangent plane will be used for calculating the rigid motion invariant;

(3) by the relation in section and the secondary section of original mesh model mid point, illustrate that there is local frame in the differential zone of putting on the curved surface, the point in this zone distributes in this part frame;

(4) from original mesh model any point can be expressed to the vector that the consecutive point of this point send in the position of any point in the original mesh model in the known original mesh model, determines section and the normal vector of this point;

(5) section direction and the normal vector direction of any point in the original mesh model curved surface are carried out parametrization, calculate successively the projection of vector on the section that a little consists of to its all consecutive point, the angle between the length of this projection and the consecutive point projection belongs to the discrete magnitude of tangent plane direction; Point belongs to the discrete magnitude on the normal direction to the projection of vector on the normal vector direction that consecutive point consist of; Discrete magnitude on tangent plane direction and the normal direction is to choose with the parameter of this point and rigid motion invariant that the coordinate system of this point all has nothing to do, and these discrete magnitudes are called discrete basic parameter.

Being implemented as follows of described step (2):

(1) at first calculates the elevation angle that all consecutive point of any point in the original mesh model leave this section;

(2) choose of elevation angle maximum in the consecutive point, the vector that this point and this consecutive point are consisted of is called the principal direction on this summit;

(3) for any point in the original mesh model, all are comprised the normal vector of face of this point and the area of face is done weighting as the normal vector of this point;

(4) vector that consists of with the consecutive point that point to summit principal direction, projection limit on the section is as the x axle of frame, with a while perpendicular to the vector of unit length of x axle and normal vector as the y axle, then the vector of unit length of x axle, y axle and normal vector has consisted of a right-handed scale (R.H.scale) base, is called frame of axes; With this right-handed scale (R.H.scale) base standardization, it is exactly the local frame at this some place.

Determine in the described step (3) and to find the solution the step of Discrete Surfaces equation as follows:

(1) the local frame between the consecutive point is poor also can be only represents that by the local frame of current this some the expression mode with the local frame phasor difference between this expression mode substitution consecutive point obtains two relations between the consecutive point frame of axes;

(2) for the relation of each point in the model with its consecutive point, consist of a system of equations, its unknown number is the local frame of the grid model mid point after the distortion, and this system of equations is called the Discrete Surfaces equation;

When (3) finding the solution this Discrete Surfaces system of equations, need to carry out C＋＋ language with growth equation group right-hand member vector with user interactions as binding occurrence, solving result is called new local frame a little for the local frame of the grid model mid point after being out of shape.

The present invention's beneficial effect compared with prior art is: the present invention at first extracts rigid motion invariant in the grid, sets up invariable rotary differential coordinate, so that can reach preferably deformation effect when curved surface carried out the conversion such as big angle rotary; Simultaneously can adopt interactive mode to provide constraint by the user in real time, make the user when carrying out product design, cartoon role creation, can by three-dimensional grid model is carried out deformation operation, reach the purpose that product is adjusted in real time and revised.

Description of drawings

Fig. 1 is overall process synoptic diagram of the present invention;

Fig. 2 is curved surface differential structrue synoptic diagram of the present invention;

Fig. 3 is the synoptic diagram of the local frame of point of the present invention;

Fig. 4 is the poor synoptic diagram of local frame between the arbitrary neighborhood point in the curved surface of the present invention;

Fig. 5 is embodiments of the invention effect synoptic diagram; The left side is the original mesh model, and the right side is the rear New model effect of distortion.

Embodiment

Below in conjunction with accompanying drawing and embodiment the present invention is described in further detail:

The invention process process comprises five key steps: calculate the discrete basic parameter of original mesh model curved surface, set up the local frame of point, determine and find the solution the Discrete Surfaces equation, set up point equation, the calculated deformation effect.As shown in Figure 1.

Step 1 is calculated the discrete basic parameter of original mesh model curved surface.

Differential structrue to curved surface is decomposed, for three-dimensional grid model curved surface P (u ₁, u ₂) ((u ₁, u ₂) expression curved surface bent line parameter), tangent plane to a surface and parameter are chosen irrelevant, to any point on it

Unit normal vector n is herein:

${(\frac{\∂P}{{\∂}_{u_1}}\×\frac{\∂P}{{\∂}_{u_2}}/||\frac{\∂P}{{\∂}_{u_1}}\×\frac{\∂P}{{\∂}_{u_2}}|\left|\right)}_{(u_1^0,u_2^0)}$

The section of this point Just can be determined by n that it is two parameter local derviation vectors

Linear combination.With P (u ₁, u ₂) at p ₀Near the Taylor expansion point, most end include the high-order infinitesimal of parameter secondary increment, can not considered.Then

On the upper p of the some replacement curved surface S that can be similar to ₀Near the point point claims A section for S.But by quadric surface Q of a section explication, Q is the upper p of approximate substitution curved surface S better ₀Near point; Q is called the secondary tangent plane in the projection of the method direction of S.

As shown in Figure 2, given i its position in model of point is With

(d _iThe limit number that expression is sent from i point) represents respectively from point Neighbours to it put the vector that sends.Definition of T _iM is the section of i point, N ⁱIt is the normal vector of i point. Expression

At T _iProjection on the M,

The discrete magnitude that belongs to the tangent plane direction;

Belong to the discrete magnitude on the normal direction.These discrete magnitudes are called discrete basic parameter.

Step 2, the local frame of foundation point mainly is divided into two stages:

First stage: the principal direction of finding the solution the summit

At first the elevation angle in this section is left on all neighbours summits of calculation level i.Choose elevation angle maximum in the neighbours point one, the vector that an i and this neighbours are put formation is called the principal direction on this summit.

Second stage: the local frame of structure point

For any point i in the model, all are comprised the normal vector of face of this point and the area of face is done weighting as the normal vector N of point ⁱAccording to a secondary tangent plane and three linearly independent vectors

With the contact of n, can reach a conclusion: to the upper p of curved surface S ₀The differential zone have local frame: Point that should the zone can be distributed in this frame.As shown in Figure 3, the vector that consists of with the consecutive point that point to principal direction

The x axle that is projected as frame on the section, with one perpendicular to x axle and N ⁱVector of unit length n as the y axle, then

Consisted of a right-handed scale (R.H.scale) base.This coordinate disjunction mark standard is turned to

Putting exactly i is positioned at The local frame at place.

Step 3 is determined and is found the solution the Discrete Surfaces equation, mainly is divided into two stages:

First stage: the local frame in the calculating curved surface between the arbitrary neighborhood point is poor

For two adjacent vertexs (i, j) on the limit, definition difference operator δ is $\left\{\begin{array}{c}{\mathrm{\δ}}_j\left(b_1^i\right)=b_1^j-b_1^i\\ {\mathrm{\δ}}_j\left(b_2^i\right)=b_2^j-b_2^i\\ {\mathrm{\δ}}_j\left(N^i\right)=N^j-N^i\end{array}\right.$ $\left(1\right)$

Can be by the local frame of consecutive point

Expression, and can obtain

In coordinate, as shown in Figure 4.

Second stage: definition Discrete Surfaces equation

(1) formula also can be expressed as

$\left\{\begin{array}{c}{\mathrm{\δ}}_j\left(b_1^i\right)=A_{11}{b}_1^i+A_{12}{b}_2^i+A_{13}{N}^i\\ {\mathrm{\δ}}_j\left(b_2^i\right)=A_{21}{b}_1^i+A_{22}{b}_2^i+A_{23}{N}^i\\ {\mathrm{\δ}}_j\left(N^i\right)=A_{31}{b}_1^i+A_{32}{b}_2^i+A_{33}{N}^i\end{array}\right.---\left(2\right)$

Formula (1) substitution formula (2) can be got formula

$\left\{\begin{array}{c}{b}_1^j=(A_{11}+1)b_1^i+A_{12}{b}_2^i+A_{13}{N}^i\\ b_2^j=A_{21}{b}_1^i+(A_{22}+1)b_2^i+A_{23}{N}^i\\ N^j=A_{31}{b}_1^i+A_{32}{b}_2^i+(A_{33}+1)N^i\end{array}\right.---\left(3\right)$

Wherein $\left\{\begin{array}{c}{A}_{11}+1=\mathrm{dot}(b_1^j,b_1^i)\\ A_{12}=\mathrm{dot}(b_1^j,b_2^i)\\ A_{13}=\mathrm{dot}(b_1^j,N^i)\\ A_{21}=\mathrm{dot}(b_2^j,b_1^i)\\ A_{22}+1=\mathrm{dot}(b_2^j,b_2^i)\\ A_{23}=\mathrm{dot}(b_2^j,N^i)\\ A_{31}=\mathrm{dot}(N^j,b_1^i)\\ A_{32}=\mathrm{dot}(N^j,b_2^i)\\ A_{33}+1=\mathrm{dot}(N^j,N^i)\end{array}\right.,$ Expression be dot product between vector.For each point in the model and its consecutive point relation, formula (3) can consist of a system of equations, and its unknown number is the new local frame of each point

This system of equations is called the Discrete Surfaces equation.

When finding the solution this Discrete Surfaces system of equations, need to user interactions as binding occurrence, carry out C＋＋ language with growth equation group right-hand member vector.

Step 4: set up point equation.

Solve in each vertex v _i' locate new local frame

Regard the discrete basic parameter of point as v _i' the coordinate of vector in new local frame that consist of to its consecutive point.There is such relation between new consecutive point:

${\hat{x}}^{j\&prime;}-{\hat{x}}^{i\&prime;}=<{\tilde{x}}_k^i,b_1^i>b_1^{i\&prime;}+<{\tilde{x}}_k^i,b_2^i>b_2^{i\&prime;}+{\tilde{L}}_k^i{N}^{i\&prime;---\left(4\right)}$

Wherein

The expression vertex v _i' the coordinate of k consecutive point after distortion.Formula (4) can consist of a system of equations, and its unknown number is the coordinate of point after the distortion.This system of equations is called point equation.

Step 5: calculated deformation effect

Customer interaction information as binding occurrence, is generated point equation group right-hand member vector and carries out C＋＋ language; Result of calculation is the position of the rear grid model mid point of distortion; The interactive information different by the user can obtain different deformation effect.The user can provide certain binding occurrence by three-dimensional grid model is carried out deformation operation when carrying out product design, cartoon role creation, reach the purpose that product is adjusted in real time and revised.As shown in Figure 5, the model in the present embodiment can be by the part of user's shirtsleeve operation model, the change of implementation model; This deformation method reciprocal process simple, intuitive, the minutia of model is kept after the distortion.

Claims (4)

1. alternately geometry deformation method based on the constant differential coordinate of Linear Rotation is characterized in that step is as follows:

(1) calculates the discrete basic parameter of original mesh model curved surface

Differential structrue to original mesh model curved surface is decomposed, and extracts the rigid motion invariant that the coordinate system choosing and put with the parameter of putting all has nothing to do, and is called discrete basic parameter; Described discrete basic parameter will be as local frame and the point equation of setting up the grid model mid point;

(2) the local frame of foundation point

For any one point in the original mesh model, the principal direction according to normal vector and this point of this point constructs the right-handed Cartesian coordinate system that initial point places this point, is called local frame a little; Local frame phasor difference in the computing grid model curved surface between any two consecutive point, the local frame that obtains between the consecutive point is poor;

(3) determine and find the solution the Discrete Surfaces equation

Poor according to the local frame between the arbitrary neighborhood point in the grid model curved surface, determine the Discrete Surfaces equation, the solving result of Discrete Surfaces equation is called new local frame a little for the local frame of the grid model mid point after being out of shape;

(4) set up point equation

For any point in the grid model, with the discrete basic parameter in the original mesh model of obtaining in the step (1), regard the coordinate of vector in the new local frame of this point that this consists of to these consecutive point as; Relation between the consecutive point consists of a system of equations, and the unknown number of system of equations is the coordinate of point after the distortion, and this system of equations is called point equation;

(5) calculated deformation effect

Customer interaction information as binding occurrence, is generated point equation group right-hand member vector and carries out C＋＋ language; Result of calculation is the position of the rear grid model mid point of distortion; Adopt interactive mode to provide constraint by the user in real time, make the user when carrying out product design, cartoon role creation, by three-dimensional grid model is carried out deformation operation, reach the purpose that product is adjusted in real time and revised.

2. the alternately geometry deformation based on the constant differential coordinate of Linear Rotation according to claim 1 is characterized in that: being implemented as follows of described step (1):

(1) for original mesh model curved surface, the section of any point is determined by the unit normal vector of this point in the described curved surface, the unit normal vector of this point is the linear combination of two parameter local derviation vectors, the parameter of the section of any point and this point is chosen irrelevantly in this grid model curved surface, and the section of point can be used as calculates the rigid motion invariant;

(2) with near the original mesh model curved surface Taylor expansion arbitrfary point, the Taylor form most end includes the high-order infinitesimal of parameter secondary increment, do not considered, near the point this point on the replacement curved surface that can be similar to of the point on this grid model curved surface then, the section of this point is a section of original mesh model; Determine a quadric surface by a section; Quadric surface is called the secondary tangent plane in the projection of the method direction of original mesh model, and the secondary tangent plane will be used for calculating the rigid motion invariant;

(3) by the relation in section and the secondary section of original mesh model mid point, illustrate that there is local frame in the differential zone of putting on the curved surface, the point in this zone distributes in this part frame;

(4) from original mesh model any point can be expressed to the vector that the consecutive point of this point send in the position of any point in the original mesh model in the known original mesh model, determines section and the normal vector of this point;

(5) section direction and the normal vector direction of any point in the original mesh model curved surface are carried out parametrization, calculate successively the projection of vector on the section that a little consists of to its all consecutive point, the angle between the length of this projection and the consecutive point projection belongs to the discrete magnitude of tangent plane direction; Point belongs to the discrete magnitude on the normal direction to the projection of vector on the normal vector direction that consecutive point consist of; Discrete magnitude on tangent plane direction and the normal direction is to choose with the parameter of this point and rigid motion invariant that the coordinate system of this point all has nothing to do, and these discrete magnitudes are called discrete basic parameter.

3. the alternately geometry deformation based on the constant differential coordinate of Linear Rotation according to claim 1 is characterized in that: being implemented as follows of described step (2):

(1) at first calculates the elevation angle that all consecutive point of any point in the original mesh model leave this section;

(2) choose of elevation angle maximum in the consecutive point, the vector that this point and this consecutive point are consisted of is called the principal direction on this summit;

(3) for any point in the original mesh model, all are comprised the normal vector of face of this point and the area of face is done weighting as the normal vector of this point;

(4) vector that consists of with the consecutive point that point to summit principal direction, projection limit on the section is as the x axle of frame, with a while perpendicular to the vector of unit length of x axle and normal vector as the y axle, then the vector of unit length of x axle, y axle and normal vector has consisted of a right-handed scale (R.H.scale) base, is called frame of axes; With this right-handed scale (R.H.scale) base standardization, it is exactly the local frame at this some place.

4. the alternately geometry deformation based on the constant differential coordinate of Linear Rotation according to claim 1 is characterized in that: determine in the described step (3) and to find the solution the step of Discrete Surfaces equation as follows:

(1) the local frame between the consecutive point is poor also can be only represents that by the local frame of current this some the expression mode with the local frame phasor difference between this expression mode substitution consecutive point obtains two relations between the consecutive point frame of axes;

(2) for the relation of each point in the model with its consecutive point, consist of a system of equations, its unknown number is the local frame of the grid model mid point after the distortion, and this system of equations is called the Discrete Surfaces equation;

When (3) finding the solution this Discrete Surfaces system of equations, need to carry out C＋＋ language with growth equation group right-hand member vector with user interactions as binding occurrence, solving result is called new local frame a little for the local frame of the grid model mid point after being out of shape.

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