Ray Tracing CPSC 453 Fall 2018 Sonny Chan Ray Tracing A method - - PowerPoint PPT Presentation

Ray Tracing

CPSC 453 – Fall 2018 Sonny Chan

Ray Tracing

A method for synthesizing images

• f virtual 3D scenes.

Image Capture Devices

Which one shall we use?

Goal: Simulate a Camera Obscura!

“Spheres & Checkerboard”

Turner Whitted, 1979

Whitted Ray Tracing

• “An improved illumination

model for shaded display”

• T. Whitted, SIGGRAPH 1979
• 512 × 512 image
• Rendered on VAX 11/780
• 74 minutes compute time
• How fast would your

computer render this today?

[image courtesy of P . Hanrahan, Stanford University]

The Ray Tracing Algorithm

• For every pixel or position on your image, do
• 1. ray generation: where am I looking?
• 2. ray intersection: do I see something?
• 3. shading: what colour is it?

Ray Generation

virtual image plane

[from photojojo.com]

Let’s see how we might

generate those rays

Ray Generation

r(t) = e + t(s − e)

Ray Intersection

What do we see?

Ray Intersection

• For every object we have in our scene, we need a way to

determine whether or not we can see it

• Usually amounts to solving an equation of one variable
• We will examine three key intersection tests today:
• spheres
• planes
• triangles

spheres

Ray-Sphere Intersection

||p − c|| = R (p − c) · (p − c) − R2 = 0 r(t) = o + td (r(t) − c) · (r(t) − c) − R2 = 0 (o + td − c) · (o + td − c) − R2 = 0 A ray: A sphere: Intersection:

Solve for t…

• d

c R

planes

Ray-Plane Intersection

A ray: A plane: Intersection: r(t) = o + td (p − q) · ˆ n = 0

Solve for t…

(r(t) − q) · ˆ n = 0 (o + td − q) · ˆ n = 0

• d

q ˆ n

triangles

How might we test

intersection of a ray and a triangle?

Barycentric Coordinates

p0 p1 p2 f(u, v) = (1 − u − v)p0 + up1 + vp2 u

v

w

(.6, .4, 0) (.3, .2, .5)

Another Interpretation

Ratio of (signed) areas of triangles:

u = A1 A v = A2 A

f(u, v) = (1 − u − v)p0 + up1 + vp2

A = 1

2(p1 − p0) × (p2 − p0)

p0 p1

(.3, .2, .5)

p2

A0 A1 A2

A Direct Approach for Intersection

r(t) = o + td  −d p1 − p0 p2 − p0     t u v   = o − p0

A ray: A triangle: Rearrange terms:

f(u, v) = (1 − u − v)p0 + up1 + vp2

• + td = (1 − u − v)p0 + up1 + vp2

Ray-triangle intersect:

Solve for t, u, and v…

Cramer’s Rule

• Given a set of linear equations in matrix form
• Write the determinant of the matrix
• Then the solutions are

 a b c     x y z   = d det(a, b, c) =

• a1

b1 c1 a2 b2 c2 a3 b3 c3

• x = det(d, b, c)

det(a, b, c) y = det(a, d, c) det(a, b, c) z = det(a, b, d) det(a, b, c)

A Direct Approach for Intersection

 −d p1 − p0 p2 − p0     t u v   = o − p0 e1 = p1 − p0, e2 = p2 − p0, s = o − p0   t u v   = 1 det(−d, e1, e2)   det(s, e1, e2) det(−d, s, e2) det(−d, e1, s)  

Our equation: Applying Cramer’s rule: where

Check that t > 0 and u, v, u+v are within [0,1] interval!

Ray-Object Intersection Tests

• Now we know how to draw spheres, planes, and

triangles in perspective with a ray tracer!

• What else can we do?

[from photojojo.com]

I think we need some shading…

What we’ve got now… and what we really want!

Shadows & More

What happens if we have more than one light?

How do we get these nice, soft shadows?

Depth of Field

Can we ray trace it?

• bject

image

The Ray Tracing Algorithm

• For every pixel position on your image:
• 1. Generate a parametric viewing ray.
• 2. Test for intersection against each object in your

scene.

• What happens if it intersections more than one object?
• 3. Determine the colour to be carried back on the ray
• Trace shadow ray(s) to obtain shadow effect

Things to Remember

• Ray tracing is a relatively simple way for us to synthesize

images of a 3D scene in perspective

• simulates the optics of a pinhole camera
• We need to devise a ray-object intersection for each kind
• f object geometry in the scene
• Some effects like shadows are relatively easy to compute
• Soft shadows and depth of field are possible if you’re

willing to wait!