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Steve Marschner CS 4620 Cornell University
Steve Marschner • Cornell CS4620 Fall 2020
Ray Intersection
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- 1. Ray-sphere intersection
Steve Marschner • Cornell CS4620 Fall 2020
Ray Intersection
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Ray Intersection Steve Marschner CS 4620 Cornell University - - PowerPoint PPT Presentation
Ray Intersection Steve Marschner CS 4620 Cornell University - - PowerPoint PPT Presentation
Ray Intersection Steve Marschner CS 4620 Cornell University Cornell CS4620 Fall 2020 Steve Marschner 1 Ray Intersection 1. Ray-sphere intersection Cornell CS4620 Fall 2020 Steve Marschner 2 Ray: a half line Standard
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- Standard representation: origin point and direction
– this is a parametric equation for the line – lets us directly generate the points on the line – if we restrict to then we have a ray – note replacing with doesn’t change ray (for )
p d r(t) = p + td t > 0 d αd α > 0
Steve Marschner • Cornell CS4620 Fall 2020
Ray: a half line
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- Condition 1: point is on ray
- Condition 2: point is on sphere
– assume unit sphere; see book or notes for general
- Substitute:
– this is a quadratic equation in t
Steve Marschner • Cornell CS4620 Fall 2020
Ray-sphere intersection: algebraic
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- Solution for t by quadratic formula:
– simpler form holds when d is a unit vector but we won’t assume this in practice (reason later) – I’ll use the unit-vector form to make the geometric interpretation
Steve Marschner • Cornell CS4620 Fall 2020
Ray-sphere intersection: algebraic
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Steve Marschner • Cornell CS4620 Fall 2020
Ray-sphere intersection: geometric
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- 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); hitSurface, t = s.intersect(ray, 0, +inf) if hitSurface is not null image.set(ix, iy, white); }
Steve Marschner • Cornell CS4620 Fall 2020
Image so far
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- 2. Ray-triangle intersection
Steve Marschner • Cornell CS4620 Fall 2020
Ray Intersection
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- A coordinate system for triangles
– algebraic viewpoint: – geometric viewpoint (areas):
- Triangle interior test:
[Shirley 2000] Steve Marschner • Cornell CS4620 Fall 2020
Barycentric coordinates
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- A coordinate system for triangles
– geometric viewpoint: distances – linear viewpoint: basis of edges
Steve Marschner • Cornell CS4620 Fall 2020
Barycentric coordinates
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- Linear viewpoint: basis for the plane
– in this view, the triangle interior test is just
[Shirley 2000] Steve Marschner • Cornell CS4620 Fall 2020
Barycentric coordinates
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- Every point on the plane can be written in the form:
for some numbers β and .
- If the point is also on the ray then it is
for some number t.
- Set them equal: 3 linear equations in 3 variables
…solve them to get t, β, and all at once!
Steve Marschner • Cornell CS4620 Fall 2020
Barycentric ray-triangle intersection
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p + td a + β(b − a) + γ(c − a) p + td = a + β(b − a) + γ(c − a)
γ γ
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p + td = a + β(b − a) + γ(c − a) β(a − b) + γ(a − c) + td = a − p ⇥a − b a − c d⇤ 2 4 β γ t 3 5 = ⇥a − p⇤ 2 4 xa − xb xa − xc xd ya − yb ya − yc yd za − zb za − zc zd 3 5 2 4 β γ t 3 5 = 2 4 xa − xp ya − yp za − zp 3 5
Steve Marschner • Cornell CS4620 Fall 2020
Barycentric ray-triangle intersection
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Cramer’s rule is a good fast way to solve this system (see text Ch. 2 and Ch. 4 for details)
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- 3. Ray intersection software
Steve Marschner • Cornell CS4620 Fall 2020
Ray Intersection
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- Rays are a useful datatype: pair origin with direction
- Also very useful to make rays into ray segments
– store a start (minimum ) and end (maximum ) – ray intersections only count if is between start and end
t t t
Steve Marschner • Cornell CS4620 Fall 2020
Rays
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class Ray { Point origin Vector direction float start float end; }
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- All surfaces need to be able to intersect rays with themselves
– surfaces with multiple intersections must return the first one – convenient to return when there is no intersection
t = +∞
Steve Marschner • Cornell CS4620 Fall 2020
Surfaces
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class Surface { boolean intersect(Hit hit, Ray r) … } is there any intersection? information about first intersection ray to be intersected class Hit { float t Surface surface Vector position Vector normal … }
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Intersection with ray segments
Steve Marschner • Cornell CS4620 Fall 2020 17
[ ] t = 0 tstart tend [ ] [ ] [ ]
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- Scenes contain many objects
- Need to find the first intersection along the ray
– that is, the one with the smallest positive t value in [start, end]
Steve Marschner • Cornell CS4620 Fall 2020
Scenes
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class Scene { List<Surface> surfaces … }
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- Input is a ray and a collection of surfaces
- Simple linear time algorithm:
- This is fine for small scenes, too slow for large ones
– real applications use sublinear methods (acceleration structures) which we will see later
Steve Marschner • Cornell CS4620 Fall 2020
Intersection against many shapes
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