CONTENTS

• Perspective projections
• Problems with perspective projections
• Example of perspective projections
• Types of perspective projections

Perspective projections

Perspective projection is the principle used by artists for drawing 3D objects and scenes [1]. The plane containing the canvas is the view plane, with view plane normal . Points on an object are projected to the view plane along lines (projectors) that converge to a center of projection . The artist's eye is at the center of projection. The image produced by perspective projections are similar to that produced by a camera (or cyclops).

Let Ax + By + Cz + D = 0 be the projection plane. Let be the point to be projected. Parametric equation of the line from to

Substitute line into plane and solve for t

Note denominator for t depends on object point

Assume z=0 is the projection plane and is the center of projection. From previous equations, , and

Or by similar triangles

Problems with perspective projections

1. Perspective foreshortening: Distant lines are foreshortened. For example, in the figure, the projection of both objects A and B are of same size. Objects that are away from center of projection appears smaller.
2. The perspective projection of objects behind the center of projection appear upside down and backward onto the viewplane.
3. Relative dimensions of the objects are not preserved and hence the information destroyed.
4. Lines parallel to view plane, i.e. perpendicular to viewplane normal are projected as parallel lines. However, lines that are not parallel to viewplane or receding parallel lines appear to meet at some point on the view plane called vanishing point.

   Consider a cube oriented in such a way that the edges recede from the viewer. The perspective projection of the edges will meet at a vanishing point . A classic example is the illusion that railroad tracks meet at a point on the horizon. Similarly, the parallel lines appear to be bent as they converge to a vanishing point.

Example of perspective projections

• Let the object point be
• Project onto the z=0 plane
• Let center of projection be located at
• By similar triangles

  

  and thus

  

Various types of perspective projections

Vanishing points play a major role in perspective projection. They are used to highlight features or to increase dramatic effect in perspective drawings. Different types of perspective projections viz. 1-point, 2-point, 3-point perspective are named after the number of finite vanishing points.

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