Analytic Geometry in 3D

Analytic Geometry in 3D
An Introduction to 3D Shapes

Prof. David Bernstein
James Madison University

Computer Science Department

bernstdh@jmu.edu

Points

In Cartesian Space:
- A point in $\mathbb{R}^{3}$ is an ordered triple of numbers (called coordinates)
In Homogeneous Space:
- A point is specified using four coordinates

Points (cont.)

From Cartesian to Homogeneous:
- The Cartesian point $(x,y,z) \in \mathbb{R}^{3}$ becomes $(x,y,z,1)$
- The Cartesian point $(x,y) \in \mathbb{R}^{2}$ becomes $(x,y,0,1)$
From Homogeneous to Cartesian:
- The homogeneous point $(x,y,z,w)$ becomes the Cartesian point $(x/w,y/w,z/w)$ if $w$ is nonzero

Points (cont.)

Homogeneous Points without a Corresponding Cartesian Point:
- When $w = 0$ the point is often said to be "at infinity"
Limits:
- Consider the sequence of points $(1,2,0,1.000)$ , $(1,2,0,0.100)$ , $(1,2,0,0.010)$ , $(1,2,0,0.001)$ ,...
- This sequence corresponds to the sequence of Cartesian points $(1,2,0)$ , $(10,20,0)$ , $(100,200,0)$ , $(1000,2000,0)$ ,...
- This is a sequence of points along the line $y = 2x$

Lines

Lines (cont.)

Parametric Form:
- Given $\bs{p} \in \mathbb{R}^{3}$ and $\bs{v} \in \mathbb{R}^{3}$ , the parametric form of a line is the set of points given by $\bs{p} + \lambda \bs{v}$
Using Two Points:
- As in 2D, we can use $\bs{v} = \bs{q} - \bs{p}$ for the line between $\bs{p}$ and $\bs{q}$

Lines (cont)

In 2D:
- Two lines either intersect or are parallel
In 3D:
- Can be skew (i.e., neither intersect nor are parallel)
- Finding the itersection involves the solution of a linear system with two equations and three unknowns

Planes

Definition:
- Given a point $\bs{p} \in \mathbb{R}^{3}$ and a direction vector $\bs{n} \in \mathbb{R}^{3}$ through $\bs{p}$ , the set of points $x \in \mathbb{R}^{3}$ satisfying $\bs{n} \cdot (\bs{x} - \bs{p}) = 0$ is called the plane defined by $\bs{p}$ and $\bs{n}$
Terminology:
- This is called the implicit form of the plane
- If $|| \bs{n} || = 1$ then $\bs{n}$ is called the normal to the plane

Planes (cont.)

Parametric Form:
- Given a point $\bs{p} \in \mathbb{R}^{3}$ and two direction vectors $\bs{v} \mbox{ and } \bs{w}$ , the parametric form of a plane is given by $\bs{p} + \lambda \bs{v} + \mu \bs{w}$
Using Three Points:
- Set $\bs{v} = \bs{q} - \bs{p}$ and $\bs{w} = \bs{r} - \bs{p}$

Planar Shapes

Triangles:
- Are planar (i.e., lie in a single plane)
Other Shapes:
- Shapes formed from four or more vertices may not be planar (e.g., consider a folded piece of paper)

Halfspaces

Polyhedra

Definition:
- The intersection of a finite number of halfspaces
Examples:
- Tetrahedron (4 equilateral triangle faces)
- Pentahedron (5 faces i.e., the square pyramid and the triangular prism)

Spheres

Definition:
- The set of all points within a given radius, $r$ , of a point, $\bs{c}$
Implicit Form:
- $\{ \bs{p} \in \mathbb{R}^{3}: (\bs{p} - \bs{c}) \cdot (\bs{p} - \bs{c}) - r^2 \leq 0\}$