ray polygon intersection. lecture #15cs410/yr2013fa/more_progress/l15_raypolygon.pdfray polygon...

Ray Polygon Intersection. Lecture #15 Thursday, October 17th, 2013

Review - Ray Casting

§  Goal: throw rays through pixels, intersect them with surfaces

§  Compute surface reflectance at points of intersection

10/22/13 © Bruce Draper & J. Ross Beveridge 2012 2

PRP (fp)

Review - Cameras & Ray Casting

Given PRP, then where is pixel Px,y? §  Remember the camera’s rotation matrix:

§ N is the vector in the direction of the VPN § U is the camera’s x axis in world coordinates §  V is the camera’s y axis in world coordinates § N, U, V are all unit length

§  Focal length d is measured is from PRP to image plane


Px,y = PRP − dVPN +Ux +Vy

Review - Cameras (II)

§  What is a ray? A parametric line for t > 0. §  What ray associated with pixel Px,y? §  Move away from FP (same as PRP). § … in direction of vector with base at FP § … and tip at Px,y


Rx,y t( ) = FP + t(Px,y −FP)Rx,y t( ) = FP + t Ux +Vy− dVPN( )

Review Ray Casting (main loop)

§  A ray casting program throws a ray from the focal point through every pixel §  This is the main loop of your ray tracer

§  For each ray: §  Intersect the ray with every polygonal surface

§  Find the first point of intersection.

§ Compute the illumination at that point.

§ Use this value to fill the pixel


Ray/Surface Intersection

§  Implicit surfaces are defined by

§  Given a ray

§  The intersection is solved by


f (p) = 0

L + tU

f L + tU( ) = 0

Intersect a Polygon Face

§  Find intersection point P on infinite plane. §  Test if point P is inside polygon. §  General equation for a plane in 3D: §  Recall an (x,y,z) point P is on a plane iff:


ax + by+ cz+ d = 0

N ⋅P = −d where N = a b c

Here is a worked Example


Planar Intersection


N ⋅ L + tU( ) = −dtN ⋅U = − N ⋅L + d( )

t =− N ⋅L + d( )

N ⋅U

Be careful. What if this is zero?

Cost: 6 mults, 1 div, 5 adds, 1 neg

Inside/Outside

§  Finding point of intersection arguably easy. §  Is the point inside the bounded polygon? §  Multiple ways to approach this question

§ Odd/even parity for general polygons § Divide convex polygons into triangles, perform

triangle inside/outside test


Arbitrary polygons

Step 1: project from 3D to 2D

§  Polygon edges are vectors in 3D §  Point of ray/plane intersection is in 3D

§ … but they all reside on the same plane

§ … test inside/outside in 2D, not 3D

§  Not just faster; fewer stability issues §  How do we project onto a 2D plane?


But, there is a faster way.

§  Orthographic projection onto either: §  XY, YZ or XZ plane. § How? Just drop one dimension (set it to zero).

§  But, be careful §  If your polygon is in the XY plane (z=const) § … and you drop X or Y, § … your polygon collapses to a line.

§  Close to parallel is essentially just as bad. §  If your polygon is almost in the XY plane

(Δz≈0), then round-off can create problems 10/22/13 © Bruce Draper & J. Ross Beveridge 2012 12

Illustration – favored choice

§  Six sided polygon nearly parallel with ? §  See the trap.


XY Plane View (red,green)

XZ Plane View (red,blue)

YZ Plane View (green,blue)

Projections for Intersection (II)

§  To be safe, drop the dimension with the largest value in N (the plane normal) §  This is the coordinate most orthogonal to the

plane, and therefore the safest to drop §  Alternative: rotate the coordinate system to

make N the Z axis § Rotation matrix is easy to compute § More multiplies, but no round-off issues §  Fewer cases in your code


Rotate – align N with Z

§  Equation of the plane.

§  Rotation


N̂ ⋅P = −d where N̂ = a b c and N̂ ⋅ N̂ =1

R =? ? ?? ? ?a b c

There are a variety of ways to selection the other to orthogonal basis vectors.

2D Polygon Membership

§  Either way,

§  We now have a 2D problem: §  Polygon specified with 2D vertices

§  Point P (of planar intersection) is a 2D point

§  Task: §  is P inside or outside of the polygon?


Odd/Even Parity Rule

§  Tests whether a 2D point P is inside or outside of a polygon

§  Step 1: draw a ray from P in any direction in the plane. (we overwork word ‘ray’ here)

§  Step 2: count how many boundaries are crossed § Odd # of intersections ⇒ inside §  Even # of intersections ⇒ outside

10/22/13 17 © Bruce Draper & J. Ross Beveridge 2012

Odd/Even Illustrated

Direction doesn’t matter!

10/22/13 18 © Bruce Draper & J. Ross Beveridge 2012

Odd/Even Intersections

§  Represent boundaries as rays §  Bi = Vi+1 + tb(Vi+1 – Vi)

§  Start ray at intersection P in any direction § R = P + tp(1,0) works nicely…

§  To intersect ray R with polygon P §  Intersect R with every boundary §  An intersection is valid iff

§  tp ≥ 0 and 0 ≤ tb < 1.0

§ Odd # intersections => inside, even outside.


Is this efficient?

§  No! §  But it’s easy J § … and anyone in graphics ought to know it. § … because it is general, works for § … non-convex polygons, § … self-intersection polygons.

§  And efficiency often matters §  There are better methods for special cases…


Ray/Triangle Intersections

§  Ray/Triangle intersections are efficient and can be computed directly in 3D

§  They rely on the following implicit definition of a triangle:


P = A+β B− A( )+γ C − A( )β > 0,γ > 0,β +γ <1

Implicit Triangles


A C-A

β = 0.5 γ = 0.48

Solve for implicit intersections

§  To find intersection, f(L+tU) = 0 (slide #4) §  This is a set of 3 linear equations with 3

unknowns:


L + tU = A+β B− A( )+γ (C − A)

Lx + tUx = Ax +β Bx − Ax( )+γ Cx − Ax( )Ly + tUy = Ay +β By − Ay( )+γ Cy − Ay( )Lz + tUz = Az +β Bz − Az( )+γ Cz − Az( )

unkowns t, β, γ

Push it through …

§  Constants on one side. §  Now in Matrix Form


Lx − Ax = β Bx − Ax( )+γ Cx − Ax( )− tUx

Ly − Ay = β By − Ay( )+γ Cy − Ay( )− tUy

Lz − Az = β Bz − Az( )+γ Cz − Az( )− tUz

Bx − Ax( ) Cx − Ax( ) −Ux

By − Ay( ) Cy − Ay( ) −Uy

Bz − Az( ) Cz − Az( ) −Uz

β

γ

t=

Lx − Ax

Ly − Ay

Lz − Az

ray polygon intersection. lecture #15cs410/yr2013fa/more_progress/l15_raypolygon.pdfray polygon...

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