CN105160701A - Triangle intersection detection method applied to remote-operation 3D visual simulation - Google Patents

Abstract

The invention discloses a triangle intersection detection method applied to remote-operation 3D visual simulation. A novel triangle intersection test algorithm is provided for fine bump detection of a 3D figure model. Compared with a traditional triangle intersection test algorithm, the step of exchanging two triangles for repeated calculation is replaced, the calculation result of the same triangle is used repeatedly to reduce the calculation load, and the speed of fine bump detection for the 3D figure model is improved.

Description

A kind of triangle intersect detection method being applied to remote operating three dimensional visual simulation

Technical field

The invention belongs to the three-dimensional artificial display field of virtual reality and Space teleoperation, relate to a kind of figure collision checking method, be specifically related to a kind of triangle intersect detection method being applied to remote operating three dimensional visual simulation.

Background technology

Remote operating plays an important role in Spatial powers combining field, and it is spatial developing tendency that usage space robot completes all kinds of space tasks, and the technology such as obstacle avoidance have epochmaking application prospect.But due to the existence of time delay, the feedback informations such as delayed vision, power feel cannot provide real-time telepresenc for operator, easily mislead the operation behavior of operator, and then affect the Secure execution of remote operating task.

Teleoperation sixty-four dollar question overcomes time delay, the telepresenc of lifting operation person, therefore adopts predictive simulation technology by the forecast model of operand by three dimensional graph display out, operator and forecast model mutual.Owing to not having time delay between operator and Virtual Space robot model, therefore Virtual Space robot model can make an immediate response the input of operator, compensate for the impact of time delay on operator.

But, ensure operator and the mutual real-time of forecast model, depend on the speed that model emulation calculates, and figure prediction part the most consuming time is collision detection, when model needs to carry out meticulous collision detection, collision detection is consuming time longer, and meticulous collision detection requires to carry out test for intersection to every diabolo dough sheet of model.

Summary of the invention

The object of the invention is to solve above-mentioned the problems of the prior art, a kind of triangle intersect detection method being applied to remote operating three dimensional visual simulation is provided.

To achieve these goals, the present invention is achieved by the following technical solutions:

Be applied to a triangle intersect detection method for remote operating three dimensional visual simulation, if two three-dimensional arbitrary triangle Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃, its summit P ₁, P ₂, P ₃and Q ₁, Q ₂, Q ₃be sequence counter-clockwise, wherein, triangle Δ P ₁p ₂p ₃the triangle surface of model is grabbed, Δ Q for surrounding hand ₁q ₂q ₃grab the triangle surface of the forecast model of model manipulation object for surrounding hand, concrete detection method comprises the following steps:

Step one: set up coordinates computed system

With Δ P ₁p ₂p ₃summit P ₁for initial point, p ₁with p ₂be respectively two base vectors, set up x ^*y ^*plane, wherein p ₁=P ₂-P ₁, p ₂=P ₃-P ₁; p ₁, p ₂and normal vector p ₁× p ₂form coordinate system x ^*y ^*z ^*three base vectors, calculate [p ₁p ₂p ₁× p ₂] inverse matrix be [p ₁p ₂p ₁× p ₂] ^-1, the triangle surface of traversal forecast model;

Step 2: apex coordinate converts

If Δ Q ₁q ₂q ₃summit Q _iat coordinate system x ^*y ^*z ^*under coordinate be Q _i ^*, then:

$Q_i^{*}=\left[\begin{array}{c}{Q}_{{\mathrm{ix}}^{*}}\\ Q_{{\mathrm{iy}}^{*}}\\ Q_{{\mathrm{iz}}^{*}}\end{array}\right]={\left[\begin{array}{ccc}{p}_1& p_2& p_1\×p_2\end{array}\right]}^{-1}(\left[\begin{array}{c}{Q}_{ix}\\ Q_{iy}\\ Q_{iz}\end{array}\right]-P_1)$

Wherein 1≤i≤3 represent three summits, Q _ix, Q _iy, Q _izrepresent Q _ithree components, represent Q _i ^*three components;

Step 3: judge Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃the position relationship of place plane

Judge Q _i ^*z ^*component , wherein 1≤i≤3:

If three values are 0, then Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃coplanar, return the coplanar mark of two triangle, continue the triangle surface of traversal object module, perform step 2;

If three values equal jack per line, then Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃non-intersect, return the non-intersect mark of two triangle, continue the triangle surface of traversal object module, perform step 2;

If a value is only had to be 0 and its residual value jack per line, then Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃non-intersect, return the non-intersect mark of two triangle, continue the triangle surface of traversal object module, perform step 2;

If two values are had to be 0, then Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃non-intersect, return the non-intersect mark of two triangle, continue the triangle surface of traversal object module, perform step 2;

If there are two values contrary sign, then Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃place Plane intersects, performs step 4;

Step 4: calculate Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃the intersection point of place plane

If symbol be different from with , make original Q ₂ ^*for Q ₁ ^*, Q ₃ ^*for Q ₂ ^*, Q ₁ ^*for Q ₃ ^*;

If symbol be different from with , make original Q ₃ ^*for Q ₁ ^*, Q ₁ ^*for Q ₂ ^*, Q ₂ ^*for Q ₃ ^*;

If Δ Q ₁q ₂q ₃article two, limit and Δ P ₁p ₂p ₃intersection point be , then

$I_1^{*}=\left(\begin{array}{c}{x}_1^{*}\\ y_1^{*}\end{array}\right)=\left(\begin{array}{c}{Q}_{1x^{*}}\\ Q_{1y^{*}}\end{array}\right)+t_1\left(\begin{array}{cc}{Q}_{2x^{*}}& -Q_{1x^{*}}\\ Q_{2y^{*}}& -Q_{1y^{*}}\end{array}\right)$

$I_2^{*}=\left(\begin{array}{c}{x}_2^{*}\\ y_2^{*}\end{array}\right)=\left(\begin{array}{c}{Q}_{1x^{*}}\\ Q_{1y^{*}}\end{array}\right)+t_2\left(\begin{array}{cc}{Q}_{3x^{*}}& -Q_{1x^{*}}\\ Q_{3y^{*}}& -Q_{1y^{*}}\end{array}\right)$

Wherein $t_1=\frac{Q_{1z^{*}}}{Q_{1z^{*}}-Q_{2z^{*}}},t_2=\frac{Q_{1z^{*}}}{Q_{1z^{*}}-Q_{3z^{*}}};$

Step 5: intersection section with Δ P ₁p ₂p ₃crossing judgement

Judge coordinate relation:

If $x_1^{*}\≤0,y_1^{*}\≤0,$ $x_1^{*}(y_2^{*}-1)\≥x_2^{*}(y_1^{*}-1),y_1^{*}(x_2^{*}-1)\≥y_2^{*}(x_1^{*}-1),x_2^{*}\≥0,y_2^{*}\≥0,$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If $x_1^{*}\&le;0,0<y_1^{*}\&le;1-x_1^{*},$ $x_1^{*}(y_2^{*}-1)\&GreaterEqual;x_2^{*}(y_1^{*}-1),x_1^{*}{y}_2^{*}\&le;x_2^{*}{y}_1^{*},x_2^{*}\&GreaterEqual;0,$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If $x_1^{*}\&le;0,y_1^{*}>1-x_1^{*},$ $x_1^{*}{y}_2^{*}\&le;x_2^{*}{y}_1^{*},y_1^{*}(x_2^{*}-1)\&le;y_2^{*}(x_1^{*}-1),x_2^{*}\&GreaterEqual;0,y_2^{*}\&le;1-x_2^{*},$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If $x_1^{*}>0,y_1^{*}\&GreaterEqual;0,y_1^{*}\&GreaterEqual;1-x_1^{*},$ $x_1^{*}(y_2^{*}-1)\&le;x_2^{*}(y_1^{*}-1),y_1^{*}(x_2^{*}-1)\&le;y_2^{*}(x_1^{*}-1),y_2^{*}\&le;1-x_2^{*},$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If $x_1^{*}>0,1-x_1^{*}\&le;y_1^{*}<0,$ $x_1^{*}(y_2^{*}-1)\&le;x_2^{*}(y_1^{*}-1),x_1^{*}{y}_2^{*}\&GreaterEqual;x_2^{*}{y}_1^{*},y_2^{*}\&le;1-x_2^{*},$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If $x_1^{*}>0,y_1^{*}<0,y_1^{*}<1-x_1^{*},$ $x_1^{*}{y}_2^{*}\&GreaterEqual;x_2^{*}{y}_1^{*},y_1^{*}(x_2^{*}-1)\&GreaterEqual;y_2^{*}(x_1^{*}-1),y_2^{*}\&GreaterEqual;0,$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If all do not meet above seven conditions, then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃non-intersect, return the non-intersect mark of two triangle, continue the triangle surface of traversal object module, perform step 2.

Compared with prior art, the present invention has following beneficial effect:

The generalized case that the present invention is directed to triangle intersect collision detection judges, from calculating the principle that in three dimensions, whether any two triangles intersect, the present invention first calculates the intersection of two triangle place planes, then judges that whether intersection and one of them triangle be crossing and simplify the more complicated judgment criterion of original algorithm.Compare traditional algorithm, there is not the step exchanging two triangle and carry out recalculating in the present invention, thus can save a large amount of evaluation work by reusing same leg-of-mutton result of calculation, improves the speed of the meticulous collision detection of three-dimensional graphical model on the whole.The present invention is directed to singularity, also possessing good solution as collision to contact with triangle but the special circumstances such as not penetrate, is the algorithm of relative maturity in three-dimensional model predictive simulation.

Accompanying drawing explanation

Fig. 1 is that the present invention sets up coordinates computed system schematic diagram;

Fig. 2 is collision detection process flow diagram of the present invention.

Embodiment

Below in conjunction with accompanying drawing, the present invention will be further described in detail.

See Fig. 1 and Fig. 2, present embodiment describes and adopt triangle intersect detection algorithm of the present invention at one in mechanical arm remote operating scene, the triangle surface of traversal paw model, performs following steps.

Step 1: set up coordinates computed system

Note triangle surface is Δ P ₁p ₂p ₃, summit P ₁, P ₂, P ₃for sequence counter-clockwise.Make p ₁=P ₂-P ₁, p ₂=P ₃-P ₁, solve [p ₁p ₂p ₁× p ₂] ^-1.

The triangle surface of traversal object module, performs following steps.

Step 2: apex coordinate converts

Note triangle surface is Δ Q ₁q ₂q ₃, summit Q ₁, Q ₂, Q ₃for sequence counter-clockwise.

Calculate $Q_i^{*}=\left[\begin{array}{c}{Q}_{{\mathrm{ix}}^{*}}\\ Q_{{\mathrm{iy}}^{*}}\\ Q_{{\mathrm{iz}}^{*}}\end{array}\right]={\left[\begin{array}{ccc}{p}_1& p_2& p_1\&times;p_2\end{array}\right]}^{-1}(\left[\begin{array}{c}{Q}_{ix}\\ Q_{iy}\\ Q_{iz}\end{array}\right]-P_1),$ Wherein 1≤i≤3 represent three summits.

Step 3: judge Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃the position relationship of place plane

Judge Q _i ^*z ^*component , wherein 1≤i≤3:

(1) if three values are 0, return the coplanar mark of two triangle, continue the triangle surface of traversal object module, perform step 2;

(2) if three equal jack per lines of value, return the non-intersect mark of two triangle, continue the triangle surface of traversal object module, perform step 2;

(3) if only there is a value to be 0 and its residual value jack per line, return the non-intersect mark of two triangle, continue the triangle surface of traversal object module, perform step 2;

(4) if there are two values to be 0, return the non-intersect mark of two triangle, continue the triangle surface of traversal object module, perform step 2;

(5) if there are two value contrary signs, carry out step 4;

Step 4: calculate Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃the intersection point of place plane

(1) if symbol be different from with , make original Q ₂ ^*for Q ₁ ^*, Q ₃ ^*for Q ₂ ^*, Q ₁ ^*for Q ₃ ^*;

(2) if symbol be different from with , make original Q ₃ ^*for Q ₁ ^*, Q ₁ ^*for Q ₂ ^*, Q ₂ ^*for Q ₃ ^*.

Order $t_1=\frac{Q_{1z^{*}}}{Q_{1z^{*}}-Q_{2z^{*}}},t_2=\frac{Q_{1z^{*}}}{Q_{1z^{*}}-Q_{3z^{*}}}$

$I_1^{*}=\left(\begin{array}{c}{x}_1^{*}\\ y_1^{*}\end{array}\right)=\left(\begin{array}{c}{Q}_{1x^{*}}\\ Q_{1y^{*}}\end{array}\right)+t_1\left(\begin{array}{cc}{Q}_{2x^{*}}& -Q_{1x^{*}}\\ Q_{2y^{*}}& -Q_{1y^{*}}\end{array}\right)$

$I_2^{*}=\left(\begin{array}{c}{x}_2^{*}\\ y_2^{*}\end{array}\right)=\left(\begin{array}{c}{Q}_{1x^{*}}\\ Q_{1y^{*}}\end{array}\right)+t_2\left(\begin{array}{cc}{Q}_{3x^{*}}& -Q_{1x^{*}}\\ Q_{3y^{*}}& -Q_{1y^{*}}\end{array}\right)$

Step 5: intersection section with Δ P ₁p ₂p ₃crossing judgement

(1) if $x_1^{*}\&le;0,y_1^{*}\&le;0,$ $x_1^{*}(y_2^{*}-1)\&GreaterEqual;x_2^{*}(y_1^{*}-1),y_1^{*}(x_2^{*}-1)\&GreaterEqual;y_2^{*}(x_1^{*}-1),x_2^{*}\&GreaterEqual;0,y_2^{*}\&GreaterEqual;0,$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

(2) if $x_1^{*}\&le;0,0<y_1^{*}\&le;1-x_1^{*},$ $x_1^{*}(y_2^{*}-1)\&GreaterEqual;x_2^{*}(y_1^{*}-1),x_1^{*}{y}_2^{*}\&le;x_2^{*}{y}_1^{*},x_2^{*}\&GreaterEqual;0,$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

(3) if $x_1^{*}\&le;0,y_1^{*}>1-x_1^{*},$ $x_1^{*}{y}_2^{*}\&le;x_2^{*}{y}_1^{*},y_1^{*}(x_2^{*}-1)\&le;y_2^{*}(x_1^{*}-1),x_2^{*}\&GreaterEqual;0,y_2^{*}\&le;1-x_2^{*},$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

(4) if $x_1^{*}>0,y_1^{*}\&GreaterEqual;0,y_1^{*}\&GreaterEqual;1-x_1^{*},$ $x_1^{*}(y_2^{*}-1)\&le;x_2^{*}(y_1^{*}-1),y_1^{*}(x_2^{*}-1)\&le;y_2^{*}(x_1^{*}-1),y_2^{*}\&le;1-x_2^{*},$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

(5) if $x_1^{*}>0,1-x_1^{*}\&le;y_1^{*}<0,$ $x_1^{*}(y_2^{*}-1)\&le;x_2^{*}(y_1^{*}-1),x_1^{*}{y}_2^{*}\&GreaterEqual;x_2^{*}{y}_1^{*},y_2^{*}\&le;1-x_2^{*},$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

(6) if $x_1^{*}>0,y_1^{*}<0,y_1^{*}<1-x_1^{*},$ $x_1^{*}{y}_2^{*}\&GreaterEqual;x_2^{*}{y}_1^{*},y_1^{*}(x_2^{*}-1)\&GreaterEqual;y_2^{*}(x_1^{*}-1),y_2^{*}\&GreaterEqual;0,$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

(7) if then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

(8) if all do not meet above seven conditions, then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃non-intersect, return the non-intersect mark of two triangle, continue the triangle surface of traversal object module, perform step 2.

Above content is only and technological thought of the present invention is described; protection scope of the present invention can not be limited with this; every technological thought proposed according to the present invention, any change that technical scheme basis is done, within the protection domain all falling into claims of the present invention.

Claims (1)

1. be applied to a triangle intersect detection method for remote operating three dimensional visual simulation, it is characterized in that, if two three-dimensional arbitrary triangle Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃, its summit P ₁, P ₂, P ₃and Q ₁, Q ₂, Q ₃be sequence counter-clockwise, wherein, triangle Δ P ₁p ₂p ₃the triangle surface of model is grabbed, Δ Q for surrounding hand ₁q ₂q ₃grab the triangle surface of the forecast model of model manipulation object for surrounding hand, concrete detection method comprises the following steps:

Step one: set up coordinates computed system

With Δ P ₁p ₂p ₃summit P ₁for initial point, p ₁with p ₂be respectively two base vectors, set up x ^*y ^*plane, wherein p ₁=P ₂-P ₁, p ₂=P ₃-P ₁; p ₁, p ₂and normal vector p ₁× p ₂form coordinate system x ^*y ^*z ^*three base vectors, calculate [p ₁p ₂p ₁× p ₂] inverse matrix be [p ₁p ₂p ₁× p ₂] ^-1, the triangle surface of traversal forecast model;

Step 2: apex coordinate converts

If Δ Q ₁q ₂q ₃summit Q _iat coordinate system x ^*y ^*z ^*under coordinate be Q _i ^*, then:

$Q_i^{*}=\left[\begin{array}{c}{Q}_{{\mathrm{ix}}^{*}}\\ Q_{{\mathrm{iy}}^{*}}\\ Q_{{\mathrm{iz}}^{*}}\end{array}\right]={\left[\begin{array}{ccc}{p}_1& p_2& p_1\&times;p_2\end{array}\right]}^{-1}(\left[\begin{array}{c}{Q}_{ix}\\ Q_{iy}\\ Q_{iz}\end{array}\right]-P_1)$

Wherein 1≤i≤3 represent three summits, Q _ix, Q _iy, Q _izrepresent Q _ithree components, represent Q _i ^*three components;

Step 3: judge Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃the position relationship of place plane

Judge Q _i ^*z ^*component wherein 1≤i≤3:

If three values are 0, then Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃coplanar, return the coplanar mark of two triangle, continue the triangle surface of traversal object module, perform step 2;

If three values equal jack per line, then Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃non-intersect, return the non-intersect mark of two triangle, continue the triangle surface of traversal object module, perform step 2;

If a value is only had to be 0 and its residual value jack per line, then Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃non-intersect, return the non-intersect mark of two triangle, continue the triangle surface of traversal object module, perform step 2;

If two values are had to be 0, then Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃non-intersect, return the non-intersect mark of two triangle, continue the triangle surface of traversal object module, perform step 2;

If there are two values contrary sign, then Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃place Plane intersects, performs step 4;

Step 4: calculate Δ Q ₁q ₂q ₃with Δ P ₁p ₂p ₃the intersection point of place plane

If symbol be different from with make original Q ₂ ^*for Q ₁ ^*, Q ₃ ^*for Q ₂ ^*, Q ₁ ^*for Q ₃ ^*;

If symbol be different from with make original Q ₃ ^*for Q ₁ ^*, Q ₁ ^*for Q ₂ ^*, Q ₂ ^*for Q ₃ ^*;

If Δ Q ₁q ₂q ₃article two, limit and Δ P ₁p ₂p ₃intersection point be then

$I_1^{*}=\left(\begin{array}{c}{x}_1^{*}\\ y_1^{*}\end{array}\right)=\left(\begin{array}{c}{Q}_{1x^{*}}\\ Q_{1y^{*}}\end{array}\right)+t_1\left(\begin{array}{cc}{Q}_{2x^{*}}& -Q_{1x^{*}}\\ Q_{2y^{*}}& -Q_{1y^{*}}\end{array}\right)$

$I_2^{*}=\left(\begin{array}{c}{x}_2^{*}\\ y_2^{*}\end{array}\right)=\left(\begin{array}{c}{Q}_{1x^{*}}\\ Q_{1y^{*}}\end{array}\right)+t_2\left(\begin{array}{cc}{Q}_{3x^{*}}& -Q_{1x^{*}}\\ Q_{3y^{*}}& -Q_{1y^{*}}\end{array}\right)$

Wherein $t_1=\frac{Q_{1z^{*}}}{Q_{1z^{*}}-Q_{2z^{*}}},t_2=\frac{Q_{1z^{*}}}{Q_{1z^{*}}-Q_{3z^{*}}};$

Step 5: intersection section with Δ P ₁p ₂p ₃crossing judgement

Judge coordinate relation:

If $x_1^{*}\&le;0,y_1^{*}\&le;0,x_1^{*}(y_2^{*}-1)\&GreaterEqual;x_2^{*}(y_1^{*}-1),y_1^{*}(x_2^{*}-1)\&GreaterEqual;y_2^{*}(x_1^{*}-1),x_2^{*}\&GreaterEqual;0,y_2^{*}\&GreaterEqual;0,$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If $x_1^{*}\&le;0,0<y_1^{*}\&le;1-x_1^{*},x_1^{*}(y_2^{*}-1)\&GreaterEqual;x_2^{*}(y_1^{*}-1),x_1^{*}{y}_2^{*}\&le;x_2^{*}{y}_1^{*},x_2^{*}\&GreaterEqual;0,$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If $x_1^{*}\&le;0,y_1^{*}>1-x_1^{*},x_1^{*}{y}_2^{*}\&le;x_2^{*}{y}_1^{*},y_1^{*}(x_2^{*}-1)\&le;y_2^{*}(x_1^{*}-1),x_2^{*}\&GreaterEqual;0,y_2^{*}\&le;1-x_2^{*},$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If $x_1^{*}>0,y_1^{*}\&GreaterEqual;0,y_1^{*}\&GreaterEqual;1-x_1^{*},x_1^{*}(y_2^{*}-1)\&le;x_2^{*}(y_1^{*}-1),y_1^{*}(x_2^{*}-1)\&le;y_2^{*}(x_1^{*}-1),y_2^{*}\&le;1-x_2^{*},$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If $x_1^{*}>0,1-x_1^{*}\&le;y_1^{*}<0,x_1^{*}(y_2^{*}-1)\&le;x_2^{*}(y_1^{*}-1),x_1^{*}{y}_2^{*}\&GreaterEqual;x_2^{*}{y}_1^{*},y_2^{*}\&le;1-x_2^{*},$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If $x_1^{*}>0,y_1^{*}<0,y_1^{*}<1-x_1^{*},x_1^{*}{y}_2^{*}\&GreaterEqual;x_2^{*}{y}_1^{*},y_1^{*}(x_2^{*}-1)\&GreaterEqual;y_2^{*}(x_1^{*}-1),y_2^{*}\&GreaterEqual;0,$ Then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃intersect, terminate to intersect and detect;

If all do not meet above seven conditions, then Δ P ₁p ₂p ₃with Δ Q ₁q ₂q ₃non-intersect, return the non-intersect mark of two triangle, continue the triangle surface of traversal object module, perform step 2.