3d照相机成像原理视景体详解

1

Projection and View Frustums

David E. Johnson

I. INTRODUCTION

In mathematics, a projection reduces an N-dimensional vector space R

N

to a subspace W. This subspace is likely

lower-dimensional than the original space. For example, a shadow is a projection of 3-space onto a 2D manifold.

A projection matrix is an N × N square matrix that defines the projection, although other projection operators are

valid. An example is the dot product of a vector with a unit vector u

x

u

y

u

x

y

u

y

u

y

l

x

l

y

=

y

l

y

)u

x

y

l

y

)u

y

= (l · u)u.

A projection matrix P has the property that

P

2

= P .

This makes intuitive sense since once the dimensionality of the original object has been reduced, projecting it

again leaves it the same. Take this example, where P is

This projection takes an object from a 3D (x, y, z) space to a 2D (x, y) space. Performing matrix multiplication





1 0 0

0 1 0

0 0 0





1 0 0

0 1 0

0 0 0





=





1 0 0

0 1 0

0 0 0





= P ,

so its projection property is verified.

Makers of maps, illustrators, architects, and engineers have developed conventions for a number of projections.

projection. Figure 1 shows a orthogonal projection of a virtual object onto the viewing plane. The box surrounding

the viewing plane and virtual space represents the viewing frustum or view frustum. The view frustum represents the

region of space that is projected onto the viewing plane. It defines the field of view of the virtual camera defining

the projection.

In OpenGL, the view frustum shape is set on the GL_PROJECTION stack and the glOrtho command creates

orthogonal projections.