Plan

• Announcements: Office hours Wed night, presentations Thursday.
• (20 min) recap depth of field; look at demos
• (5 min) building MPEGs like this one of the Moving Mobile
• (20 min) Geometry: rays intersecting triangles
• (10 min) Fog, if there's time. There are plenty of good online tutorials. Just Google for it. Some useful things to know:
  • fog color, typically gray
  • Mode is one of EXP, EXP2, and LINEAR
  • density (a positive float)
  • fog start, like near
  • fog end, like far
  • Fog manual
• (25 min) Tech of Shrek

Lecture notes on Geometry

Geometry

Geometry lets us do some cool stuff. I'd like to take you through the geometry to doing the photon torpedos in

demos/animation/Laser.py

Normals to Tilted Planes

Certain planes are easy to find the normal vector to, such as the roof of the barn. The answer is the perp vector: If you have a vector like

(a,b)

(b,-a)

(-b,a)

What's the difference between these? We'll sketch them.

Equation of a Plane

Three possibilities:

• parametric: a point and two vectors. This is convenient in some ways, but not so easy for computing intersections. This representation is nice for determining whether a point is inside a triangle, and we often care more about triangles than planes.
• implicit: this equation has some nice properties:
  • it's easy to read off the normal vector,
  • if the normal is normalized (unit length), the equation tells the signed distance of a point to the plane
  • it's easy to find intersections.

Example

Suppose we have

A = (1,2,3)
B = (3,2,5)
C = (5,3,4)

• What is the parametric equation of the plane defined by that triangle?
• What is the implicit equation of the plane defined by that triangle?
• Make sure the surface normal points out of the "front" of the plane. (How can we tell?)

Next, suppose we have

Q = (6,7,5)
r = (-1,2,1)

Q = (7, -10, 0)
r = (-1,1,1)

Intersection with Multiple Objects

To find the nearest fragment, compute all the parameters of the intersection points, and choose the smallest.

Intersections with Triangles

To find the intersection with a triangle, we first intersect with the plane, then find the parameters on the "basis" vectors v and w, then test the parameters.

• First, find the point of intersection
• Find the vector from the anchor point on the plane (the point that we used to define the plane -- there doesn't have to be anything special about this point)
• First, we have to talk about the dot product as projection.
• Find the perp vectors
• compute s and t

Code for twNearestFragment

pytw/TW.py has the code; search down for NearestFragment

Written by Scott D. Anderson
scott.anderson@acm.org

This work is licensed under a Creative Commons License.