Tracing
CS Ray 465 Lecture 3
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 1
Ray tracing idea
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 2
Ray tracing algorithm
for each pixel { compute viewing ray intersect ray with scene compute illumination at visible point put result into image }
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 3
Pinhole camera
Box with a tiny hole Image is inverted Perfect image if hole innitely small Pure geometric optics based on similar triangles No depth of eld issue
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 4
Camera Obscura
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 5
Abelardo Morell
Photographer who turns hotel room into a camera obscura (pinhole optics)
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 6
Durers Ray casting machine
Albrecht Durer, 16th century
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 8
Durers Ray casting machine
Albrecht Durer, 16th century
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 9
Plane projection in drawing
Cornell CS465 Fall 2004 Lecture 3
[CS 417 Spring 2002]
2004 Steve Marschner 10
Plane projection in photography
This is another model for what we are doing
applies more directly in realistic rendering
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 11
[CS 417 Spring 2002]
Simplied pinhole camera
Eye-image pyramid (frustum) Note that the distance/size of image are arbitrary
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 12
Generating eye rays
Use window analogy directly
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 13
Vector math review
Vectors and points Vector operations
addition scalar product
More products
dot product cross product
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 14
Ray: a half line
Standard representation: point p and direction d
this is a parametric equation for the line lets us directly generate the points on the line if we restrict to t > 0 then we have a ray note replacing d with ad doesnt change ray (a > 0)
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 15
Generating eye rays
Just need to compute the view plane point q:
we wont worry about the details for now
Cornell CS465 Fall 2004 Lecture 3 2004 Steve Marschner 16
Sphere equation
Sphere equation (implicit):
Assume unit dimensions, centered at origin
Cornell CS465 Fall 2004 Lecture 3
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Explicit vs. implicit?
Sphere equation is implicit
Solution of an equation Does not tell us how to generate a point on the sphere Tells us how to check that a point is on the sphere
Ray equation is explicit
Parametric How to generate points Harder to verify that a point is on the ray
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 18
Ray-sphere intersection: algebraic
Condition 1: point is on ray Condition 2: point is on sphere
assume unit sphere
Substitute:
this is a quadratic equation in t
Cornell CS465 Fall 2004 Lecture 3 2004 Steve Marschner 19
Ray-sphere intersection: algebraic
Solution for t by quadratic formula:
simpler form holds when d is a unit vector but we wont assume this in practice (reason later) Ill use the unit-vector form to make the geometric interpretation
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 20
Ray-sphere intersection: algebraic
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 21
Ray-sphere intersection: geometric
What geometric information is important?
Inside/outside Closest point Direction
Geometric considerations can help shortcut calculations
easy reject
O r D
Cornell CS465 Fall 2004 Lecture 3
R
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Ray-sphere intersection: geometric
Find if the rays origin is outside the sphere
R2>r2
If inside, it intersects If on the sphere, it does not intersect (avoid degeneracy)
O r D
Cornell CS465 Fall 2004 Lecture 3
R
2004 Steve Marschner 23
Ray-sphere intersection: geometric
Find if the rays origin is outside the sphere Find the closest point to the sphere center
tP=RO.D If tP<0, no hit
O r
Cornell CS465 Fall 2004 Lecture 3
P
tP
D
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Ray-sphere intersection: geometric
Find if the rays origin is outside the sphere Find the closest point to the sphere center If tP<0, no hit
Else nd squared distance d2
Pythagoras: if d2=R2-tP2 d2> r2 no hit
O
d
Cornell CS465 Fall 2004 Lecture 3
r P
tP
D
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2004 Steve Marschner 25
Ray-sphere intersection: geometric
Find if the rays origin is outside the sphere Find the closest point to the sphere center If tP<0, no hit Else nd squared distance d2 if d2 > r2 no hit
If outside t = tP-t
t2+d2=r2
If inside t = tP+t
O
d
P
Cornell CS465 Fall 2004 Lecture 3
r
t t tP
D R
2004 Steve Marschner 26
Ray-sphere intersection: geometric
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 27
Geometric vs. algebraic
Algebraic was more simple (and more generic) Geometric is more efcient
Timely tests In particular for outside and pointing away
Cornell CS465 Fall 2004 Lecture 3
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Image so far
With eye ray generation and sphere intersection
Surface s = new Sphere((0.0, 0.0, 0.0), 1.0); for 0 <= iy < ny for 0 <= ix < nx { ray = camera.getRay(ix, iy); if (s.intersect(ray, 0, +inf) < +inf) image.set(ix, iy, white); }
Cornell CS465 Fall 2004 Lecture 3
2004 Steve Marschner 29
Ray-plane intersection
Condition 1: point is on ray Condition 2: point is on plane Condition 3: point is on the inside of all edges First solve 1&2 (rayplane intersection)
substitute and solve for t:
Cornell CS417 Spring 2003 Lecture 33
2003 Steve Marschner 30
Ray-triangle intersection
In plane, triangle is the intersection of 3 half spaces
Cornell CS417 Spring 2003 Lecture 33
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Inside-edge test
Need outside vs. inside Reduce to clockwise vs. counterclockwise
vector of edge to vector to x
Use cross product to decide
Cornell CS417 Spring 2003 Lecture 33
2003 Steve Marschner 32
Ray-triangle intersection
Cornell CS417 Spring 2003 Lecture 33
2003 Steve Marschner 33
Intersection against many shapes
The basic idea is:
hit (ray, tMin, tMax) { tBest = +inf; hitSurface = null; for surface in surfaceList { t = surface.intersect(ray, tMin, tMax); if t < tBest { tBest = t; hitSurface = surface; } } return hitSurface, t; }
this is linear in the number of shapes but there are sublinear methods (acceleration structures)
Cornell CS465 Fall 2004 Lecture 3 2004 Steve Marschner 34
Image so far
With eye ray generation and scene intersection
Geometry g = new Sphere((0.0, 0.0, 0.0), 1.0); for 0 <= iy < ny for 0 <= ix < nx { ray = camera.getRay(ix, iy); c = scene.trace(ray, 0, +inf); image.set(ix, iy, c); } trace(ray, tMin, tMax) { surface, t = hit(ray, tMin, tMax); if (surface != null) return surface.color(); else return black; }
Cornell CS465 Fall 2004 Lecture 3 2004 Steve Marsc...