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# A question about polygons - PowerPoint PPT Presentation

A question about polygons. Devising a computational method for determining if a point lies in a plane convex polygon . Problem background. Recall our ‘globe.cpp’ ray-tracing demo It depicted a scene showing two objects One of the objects was a square tabletop

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### A question about polygons

Devising a computational method for determining if a point lies in a plane convex polygon

• Recall our ‘globe.cpp’ ray-tracing demo

• It depicted a scene showing two objects

• One of the objects was a square tabletop

• Its edges ran parallel to x- and z-axes

• That fact made it easy to determine if the spot where a ray hit the tabletop fell inside or outside of the square’s boundary edges

• We were lucky!

• What if we wanted to depict our globe as resting on a tabletop that wasn’t a nicely aligned square? For example, could we show a tabletop that was a hexagon?

• The only change is that the edges of this six-sided tabletop can’t all be lined up with the coordinate system axes

• We need a new approach to determining if a ray hits a spot that’s on our tabletop

• How can we tell if a point lies in a triangle?

c

q

p

b

a

Point p lies inside triangle Δabc

Point q lies outside triangle Δabc

• Draw vectors from a to b and from a to c

• We can regard these vectors as the axes for a “skewed” coordinate system

• Then every point in the triangle’s plane would have a unique pair of coordinates

• We can compute those coordinates using Cramer’s Rule (from linear algebra)

c

ap = c1*ab + c2*ac

p

b

a

c1 = det( ap, ac )/det( ab, ac )

c2 = det( ab, ap )/det( ab, ac )

y-axis

p = (c1,c2)

x-axis

x + y = 1

p lies outside the triangle if c1 < 0 or c2 < 0 or c1+c2 > 1

typedef float scalar_t;

typedef struct { scalar_t x, y; } vector_t;

scalar_t det( vector_t a, vector_t b )

{

return a.x * b.y – b.x * a.y;

}

theta = 2*PI / 6

( cos(2*theta), sin(2*theta) )

( cos(1*theta), sin(1*theta) )

( cos(3*theta), sin(3*theta) )

( cos(0*theta), sin(0*theta) )

( cos(5*theta), sin(5*theta) )

( cos(4*theta), sin(4*theta) )

A point p lies outside the hexagon -- unless

it lies inside one of these four sub-triangles

• Demo program ‘hexagon.cpp’ illustrates the use of our just-described algorithm

• Every convex polygon can be subdivided into triangles, so the same ideas can be applied to any regular n-sided polygon

• Exercise: modify the demo-program so it draws an octagon, a pentagon, a septagon

• A tetrahedron is a 3D analog of a triangle

• It has 4 vertices, located in space (but not all vertices can lie in the same plane)

• Each face of a tetrahedron is a triangle

c

b

o

a

typedef float scalar_t;

typedef struct { scalar_t x, y, z; } vector_t;

scalar_t det( vector_t a, vector_t b, vector_t c )

{

scalar_t sum = 0.0;

sum += a.x * b.y * c.z – a.x * b.z * c.y;

sum += a.y * b.z * c.x – a.y * b.x * c.z;

sum += a.z * b.x * c.y – a.z * b.y * c.x;

return sum;

}

• Let o, a, b, c be vertices of a tetrahedron

• Form the three vectors oa, ob, oc and regard them as the coordinate axes in a “skewed” 3D coordinate system

• Then any point p in space has a unique triple of coordinates: op = c1*oa + c2*ob + c3*oc

• These three coordinates can be computed using Cramer’s Rule

c1 = det( op, ob, oc )/det( oa, ob, oc )

c2 = det( oa, op, oc )/det( oa, ob, oc )

c3 = det( oa, ob, op )/det( oa, ob, oc )

Point p lies inside the tetrahedron – unless

c1 < 0 or c2 < 0 or c3 < 0

or c1 + c2 + c3 > 1

• Just as a convex polygon can be divided into subtriangles, any convex polyhedron can be divided into several tetrahedrons

• We can tell if a point lies in the polyhedron by testing to see it lies in one of the solid’s tetrahedral parts

• An example: the regular dodecahedron can be raytraced by using these ideas!