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Projecting from 3D to the Plane
An Introduction
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Prof. David Bernstein
James Madison University
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Computer Science Department
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bernstdh@jmu.edu
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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^{*}/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^{*}/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/d + 1
\end{array}\right]
- Converting to Homogeneous Coordinates:
- Divide by z/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
There's Always More to Learn
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