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:
- \(\{ \bs{p} \in \mathbb{R}^{3}: (\bs{p} - \bs{c}) \cdot (\bs{p} - \bs{c}) - r^2 \leq 0\}\)