The figure shows 2 control polygons formed by V0, V1, …, V6=V0, Connecting the v

Question

The figure shows 2 control polygons formed by V0, V1, …, V6=V0, Connecting the vertices, in the indicated order the perturbed control polygon results from changing V3 to V3', creating two triangles for examination, with their vertices listed in order as T formed by V4, V3, V3' and T' formed by V2, V3, V3'.

Problem: Show that the segments, denoted by [V0, V1], [V5, V6] do not intersect either T or T',

and that the segment [V4, V5] intersect only T, but only at V4

and that the segment [V1, V2] intersect only T', but only at V2.

Your analysis should be based upon a ray triangle intersection algorithm, using exact computation. Provide a summary of your calculations.

Explain how the same algorithm, implemented in floating point arithmetic would be insufficient to fully answer the posed question.

Explanation / Answer

Solution:- The Moller-Trumbore algorithm is a ray triangle intersection algorithm.

Let P which is some where on the ray defined by its origin O in the direction of the ray.

Intersection P can be writen as, P= 0+tD, where t is the distance from the ray's origin to the intersection P.

The plane equation is given as, Ax+By +Cz+D=0

where A,B,C are the coordinates of the normal to the plane.

D is the distance from the origin.

The variables x,y,z are the coordinates of any point which lies on the plane.

Here in the given problem T is formed with the vertices v4, v3 and v'3

and T' is formed with vertices v2,v3 and v'3

Any of these triangle's vertices lies in the plane.

In a ray triangle intersection, first we will compute the triangle's normal, then test if the ray and the triangle are parallel.IF they are parallel then the intersection test fails. If they are not parallel, then we can compute the intersection point P.

If P is on the left side of each one of the triangle's edges, then the ray intersects the triangle and P is inside the boundaries.The test is successful.

The algorithm for the ray intersects the triangle inside the boundaries is as follows

Vec3f edge0 = v3 - v4 ;

Vec3f edge1 = v'3 - v3 ;

Vec3f edge2 = v4 - v'3 ;

Vec3f C0      = P - v4 ;

Vec3f   C1      = P - v3 ;

Vec3f    C2      = P - v'3 ;

if (dotProduct ( N , CrossProduct (edge0, C0 ) ) > 0 & &

   dotProduct ( N , CrossProduct (edge1 , C1 ) ) > 0 & &

Thes same algorithm is used for the vertices v2,v3 and v'3.

then check whether the ray is inside the triagle or not,

and by using the above information we can show the segments [V0 ,V1] [V5 ,V6} do not intersect either T or T'.and the segment [V4 ,V5] intersects only T, but only at V4 and the segmet [V1 ,V2] intersects T' , but only at V2.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects