Some Classical Projections

Some Classical Projections
An Introduction

Computer Science Department

bernstdh@jmu.edu

Motivation

What We Have:
- A way to represent shapes in 3D
- 2D visual output devices
- A way to reduce the dimensionality of our shapes (i.e., projections)
What We Need:
- A way to generate "useful"/"attractive" 2D views of the 3D shapes

Approaches

Move the Projection Plane:
- Projecting onto \(z=0\) tends to not be very "useful"
- Instead we can project onto a more "useful" plane
Move the Object before Projecting:
- Orient/transform the object so that we get a more "useful" view

Approaches (cont.)

Orthographic Views

Multiview:
- The projection plane is parallel to one of the "principal faces"
- Example: An elevation (i.e., side view) in architecture
- The name comes from the fact that we need multiple views to see more than one "principal face"
Axonometric:
- We see three "principal faces" at the same time
- (Note: An axonometric transform matrix has a determinant of 0)

Axonometric Views

Trimetric:
- Three different foreshortenings
Dimetric:
- Two different foreshortenings (i.e., one for one axis and one for the other axes)
- The projection plane is (symmetrically) at an edge where two faces meet
Isometric:
- One foreshortening (i.e., all axes have the same foreshortening)
- The projection plane is (symmetrically) at a corner where three faces meet

Trimetric View

Trimetric View (cont.)

Rotation Around the \(y\)-Axis:
- \( \left[ \begin{array}{c c c c} \cos \theta & 0 & \sin \theta & 0 \\ 0 & 1 & 0 & 0 \\ - \sin \theta & 0 & \cos \theta & 0 \\ 0 & 0 & 0 & 1\end{array}\right] \)
Rotation Around the \(x\)-Axis:
- \( \left[ \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & \cos \phi & -\sin \phi & 0 \\ 0 & \sin \phi & \cos \phi & 0 \\ 0 & 0 & 0 & 1\end{array}\right] \)

Trimetric View (cont.)

Unit Vector on the \(y\)-Axis:
- \( \left[ \begin{array}{c c c c}0 \\ 1 \\ 0 \\ 1\end{array}\right] \rightarrow \left[ \begin{array}{c c c c}0 \\ \cos \phi \\ \sin \phi \\ 1\end{array}\right] \)
- Projected: \( \left[ \begin{array}{c c c c}0 \\ \cos \phi \\ 0 \\ 1\end{array}\right] \)
- Length (after projection): \( \sqrt{\cos^{2} \phi} \)
Unit Vector on the \(x\)-Axis:
- \( \left[ \begin{array}{c c c c}1 \\ 0 \\ 0 \\ 1\end{array}\right] \rightarrow \left[ \begin{array}{c c c c}\cos \theta \\ \sin \theta \sin \phi \\ -\sin \theta \cos \phi \\ 1\end{array}\right] \)
- Projected: \( \left[ \begin{array}{c c c c}\cos \theta \\ \sin \theta \sin \phi \\ 0 \\ 1\end{array}\right] \)
- Length (after projection): \( \sqrt{\cos^{2} \theta + (\sin \theta \sin \phi)^{2}} \)

Dimetric View

We Want:
- The foreshortening ratio to be the same
- That is: \( \sqrt{\cos^{2} \theta + (\sin \theta \sin \phi)^{2}} = \sqrt{\cos^{2} \phi} \)
Implications:
- \( \Rightarrow \cos^{2} \theta + (\sin \theta \sin \phi)^{2} = \cos^{2} \phi \)
- \( \Rightarrow \cos^{2} \theta + (\sin^{2} \theta \sin^{2} \phi) = \cos^{2} \phi \)
- Since \( \cos^{2} \alpha = 1 - \sin^{2} \alpha \) it follows that:
- \( \Rightarrow 1 - \sin^{2} \theta + (\sin^{2} \theta \sin^{2} \phi) = 1 - \sin^{2} \phi \)
- \( \Rightarrow - \sin^{2} \theta + (\sin^{2} \theta \sin^{2} \phi) = - \sin^{2} \phi \)
- \( \Rightarrow - \sin^{2} \theta (1 - \sin^{2} \phi) = - \sin^{2} \phi \)
- \( \Rightarrow \sin^{2} \theta = \frac{\sin^{2} \phi}{(1 - \sin^{2} \phi)} \)

Isometric View

We Want:
- All three foreshortening ratios to be the same
- That is: \( \sqrt{\cos^{2} \theta + (\sin \theta \sin \phi)^{2}} = \sqrt{\cos^{2} \phi} = \sqrt{\cos^{2} \theta} \)
After Some Algebra:
- \( \sin \theta = \sqrt{1/2} \)
- \( \sin \phi = \sqrt{1/3} \)
- So \(\phi\) must be 35.26439 degrees and \(\theta\) must be 45 degrees

Perspective Projections

Recall:
- \(x^{*} = \frac{x}{z/d +1}\)
- \(y^{*} = \frac{y}{z/d +1}\)
Questions:
- What are the signs of \(\frac{\partial x^*}{\partial x}\) and \(\frac{\partial x^*}{\partial z}\)?
- What are the signs of \(\frac{\partial y^*}{\partial y}\) and \(\frac{\partial y^*}{\partial z}\)?
- What do these signs mean for points that are farther/closer to the projection plane?
- What do these signs mean for points that are the same except for their \(x\) or \(y\) coordinates?

Perspective Projections (cont.)

Using the Visualizations to Answer these Questions

One-Point Perspective

Approach:
- Center the object
- Use a perspective transform with a vanishing point on the \(z\)-axis
Visualizing the Unit Cube:

Two-Point Perspective

Approach:
- Rotate the shape \(\theta\) around the \(y\)-axis
- Translate the shape
- Project
The Matrices:
- \( \left[ \begin{array}{c c c c} \cos \theta & 0 & \sin \theta & 0 \\ 0 & 1 & 0 & 0 \\ - \sin \theta & 0 & \cos \theta & 0 \\ 0 & 0 & 0 & 1\end{array}\right] , \left[ \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & m \\ 0 & 0 & 1 & n \\ 0 & 0 & 0 & 1\end{array}\right] , \left[ \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1/d & 1\end{array}\right] \)
The Product:
- \( \left[ \begin{array}{c c c c} \cos \theta & 0 & \sin \theta & 0 \\ 0 & 1 & 0 & m \\ 0 & 0 & 0 & 0 \\ - (\sin \theta) /d & 0 & (\cos \theta) / d & n/d + 1\end{array}\right] \)

Two-Point Perspective

Visualization

Three-Point Perspective

Approach:
- Rotate the shape \(\theta\) around the \(y\)-axis and \(\phi\) around the \(x\)-axis
- Translate the shape
- Project
- Convert back to homogeneous coordinates
The Matrices:
- \( \left[ \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & \cos \phi & -\sin \phi & 0 \\ 0 & \sin \phi & \cos \phi & 0 \\ 0 & 0 & 0 & 1\end{array}\right] \)
- \( \left[ \begin{array}{c c c c} \cos \theta & 0 & \sin \theta & 0 \\ 0 & 1 & 0 & 0 \\ - \sin \theta & 0 & \cos \theta & 0 \\ 0 & 0 & 0 & 1\end{array}\right] \)
- \( \left[ \begin{array}{c c c c} 1 & 0 & 0 & s \\ 0 & 1 & 0 & m \\ 0 & 0 & 1 & n \\ 0 & 0 & 0 & 1\end{array}\right] \)
- \( \left[ \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1/d & 1\end{array}\right] \)

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