Projecting from 3D to the Plane
An Introduction
|
|
Prof. David Bernstein
James Madison University
|
|
| Computer Science Department |
| bernstdh@jmu.edu |
|
Motivation
- What We Have:
- A way to represent shapes in 3D
- 2D visual output devices
- What We Need:
- A way to reduce the dimensionality of our shapes
Some Intuition
- The Traditional Approach:
- An Alternative Approach:
- Think about an object that produces light rays
from each point on its surface
- Each ray is called a projector
- Each ray has a direction of projection
- Each ray passes through a translucent screen (called the
projection plane)
Projection Matrix
- Defined:
- An \(n \times n\) matrix that takes a
point in \(\mathbb{R}^{n}\) to a subspace
- Properties:
- The columns of \(\bs{P}\) are the projections of the
standard basis vectors
- A matrix \(\bs{P}\) is a projection matrix iff
\( \bs{P} \bs{P} = \bs{P}\)
Orthogonal/Orthographic Projections
- Defined:
- The projectors are all parallel
- The projection plane is perpendicular to the
projectors
- A Warning About Terminology:
- A matrix is often said to be orthogonal if
\(\bs{A}^{T} \bs{A} = \bs{I}\)
- Rotation matrices and reflection matrices have
this property but are not projection matrices
Orthographic Projections (cont.)
- Projecting onto the Plane \(z = n\):
- We want to project the point
\(\left[ \begin{array}{c}x \\ y \\ z \\1 \end{array}\right]\)
- to the point \(\left[ \begin{array}{c}x \\ y \\ n \\ 1 \end{array}\right]\)
- The projection matrix is:
\(
\left[ \begin{array}{r r r r}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & n \\
0 & 0 & 0 & 1
\end{array}\right]
\)
- Projecting onto the Plane \(z = 0\):
- The projection matrix is:
\(
\left[ \begin{array}{r r r r}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
\)
Perspective Projections
- Intuition:
- All projectors meet at a (finite) Center of Projection
(COP)
- An Important Property:
- When an object is moved farther from the viewer it
gets smaller (called diminution or
foreshortening)
- From Art Class:
- Think about drawing a house and having the parallel lines
where walls meet intersecting
at a vanishing point when drawn on paper
- Which parallel lines on the object intersect depends on
whether you are using one-point, two-point, or
three-point perspective
Perspective Projections (cont.)
- A Partial Visualization:
- The Projected Point:
- \(\tan \alpha = y/(z+d)\)
- \(\tan \alpha = y^{\star}/d\)
- So:\(y^{*} = \frac{y}{z/d +1}\)
Perspective Projections (cont.)
- Another Partial Visualization:
- The Projected Point:
- \(\tan \beta = x/(z+d)\)
- \(\tan \beta = x^{\star}/d\)
- So:\(x^{*} = \frac{x}{z/d +1}\)
Perspective Projections (cont.)
- Getting Part Way:
- \(
\left[ \begin{array}{r r r r}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 1/d & 1
\end{array}\right]
\left[ \begin{array}{r}
x \\ y \\ z \\ 1
\end{array}\right]
=
\left[ \begin{array}{r}
x \\ y \\ 0 \\ z 1/d + 1
\end{array}\right]
\)
- Converting to Homogeneous Coordinates:
- Divide by \(z 1/d + 1\) to obtain:
- \(\left[\begin{array}{c}
\frac{x}{z/d +1} \\ \frac{y}{z/d +1} \\ 0 \\ 1
\end{array}\right]\)
Perspective Projections (cont.)
- One Interpretation:
- The orthographic projection is the limiting
case of the perspective projection as the COP
goes to infinity
- Another Interpretation:
- The perspective projection is the product of
a perspective transformation:
- \(
\left[ \begin{array}{r r r r}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 1/d & 1
\end{array}\right]
\)
- and an orthographic projection:
- \(
\left[ \begin{array}{r r r r}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
\)
Perspective Projections (cont.)
- Be Careful:
- Perspective projections can reverse the sign and
ordering of \(z\) values
- The Result:
- Line segments that cross \(z=0\) can be "torn"
Variants of these Projections
- One Approach:
- Think about different projection matrices
that fit in each category
- Another Approach:
- Think about orienting/transforming the object before applying
the projection
