16 3D Viewing (Projection Transformations)

T. Raghuveera

Objectives:

To Understand the Projection transformations in 3D

Discussion:

Parallel Projections:

As discussed in the previous module (Module-15), parallel projections are obtained, when objects are projected along lines that are parallel to each other. If the lines of projection are perpendicular to the plane of projection, then we get orthographic projections, otherwise we get oblique projections.Shown in the figure below are Orthographic projections.

Figure below shows an object projected along lines which make any angle other than 90° with view plane. These are oblique projections.

Orthographic parallel projections:

Here a 3D point (x,y,z) is projected as (x,y) on to a view planeas shown.Generally the view plane is assumed to be placed in the XY plane of the viewing coordinate system (whose eq. is Z=0). The default projection is orthographic projection. The shape of the view volume is a rectangular parallelepiped as shown in the figure below.

Axonometric orthographic projections:

The projection plane is aligned so that it intersects each coordinate axes in which the object is defined (principal axes) at the same distance from the origin.All the three principal axes are foreshortened equally.In other words,by aligning the viewplane normal vector along one of the principle diagonals of the cube we get isometric projections of the cube as shown in the figure below.

Oblique parallel projection:

When the lines of projection make an angle other than 90°with the view plane, we get oblique projections. Let’s assume that a point P(x,y,z) is tobe projected onto the viewplane positioned at XY plane of the VCS. The orthographic projection coordinates of the point P(x,y,z) on the view plane are (x,y), since z=0 on the view plane. Let α be the angle of projection, and the oblique projection coordinates on the view plane are say (xp,yp). Consider an imaginary line L connecting the orthogonal projection coordinates with the oblique projection coordinates as shown in the figure below. Let the line Lmake an angle ø with the horizontal / X-axis of the view plane.

The line connecting P(x,y,z) and oblique projection coordinates (xp,yp) is the oblique line of projection.The line connecting P(x,y,z) and P(x,y) is the orthogonal projection line. The cosine, sine of the angle ø, in the triangle formed by the line L and the horizontal (X-axis) is given below.

The effect of this projection matrix is to shear planes of constant z. The larger the z-value of a face / plane of an object, the more it gets sheared.An orthographic projection is obtained when L1=0.In fact the effect of the projection matrix is to shear planes of constant Z and project them on to the view plane.The angle ø is generally fixed at 30° or 60°.