Three-dimensional image projection transformation--parallel projection

(2) Parallel projection

"The projection of the sun's rays is a parallel projection."

If you move the center of the perspective projection to Infinity, each projection line becomes a straight line that is parallel to each other , and this projection is called a parallel projection.

Parallel projection can be divided into two categories according to the projection direction and the angle of the projection plane: forward projection and oblique projection

1> Positive Projection
According to the "angle" of the projection surface and the axis, it can be divided into three views and a positive axis side graph.
When the projection plane is "perpendicular" to an axis, the resulting projection is a three view, the projection direction is consistent with the direction of the axis, otherwise the resulting projection is a positive axis side view.


"1". Three views

1. Main View-->xoz face (also known as V face) as projection surface

The relationship of the point to the main view point of a three-dimensional object before and after the projection transformation matrix is:

The projection transformation matrix from the three-dimensional object to the main view is represented as:
[x ' y ' z ' 1]=[x y z 1]? Tv=[x 0 Z 1]

2. Side view-->yoz faces (also known as W faces) as projection faces

The relationship of the point to the side view point of the three-dimensional object before and after the projection transformation matrix is:

In order for the side view and the main view to be drawn in a plane, the W face is rotated 90° around the z axis, that is, there should be a rotation transformation, whose transformation matrix is:

For the main and side view to have a certain amount of space, but also to make the W face along the negative x direction of a distance-xo, the transformation matrix is:

-The projection transformation matrix for the top view is:

3. Top View-->xoy face (also called H face) is a projection surface

From the point to the top point of the three-dimensional object before and after the projection transformation, the transformation matrix is:

The projection transformation matrix from the three-dimensional object to the main view is represented as:
[x ' y ' z ' 1]=[x y z 1]? Th=[x y 0 1]

In order for the top and front view to be drawn in a plane, the H face should be rotated clockwise around the x-axis, that is, there should be a rotation transformation, whose transformation matrix is:


For the main and top view to have a certain amount of space, but also to make the H-plane in the z-direction of a distance-zo, the transformation matrix is:

-The projection transformation matrix for the top view is:

"Calculation of three views"
A. Determining the three-dimensional object"All Points"coordinates of the position;
B. Introduce the homogeneous coordinates to find out the corresponding"Transform Matrix"；
C. A matrix representation of the transformation made, by"Operations"The coordinate values of the points after the transformation of the points on the three-dimensional object are obtained.
D. Two-dimensional points obtained from the transformation"Draw out"Three-dimensional object after projection of the third view.

"Features"
A coordinate plane of an object is parallel to the projection surface,its projection can reflect the actual size of the body.。
"Poor Place"
A three-view projection of only one face of an object, so the three viewsIt is difficult to represent the three-dimensional nature of the body visually, only the main, side, and three views are put together in order to synthesize the space shape of the object.


"2". Positive axis mapping

When the projection surface is"Three axes"Between the Angles are"Equal"When the Isometric
When the projection surface is"Two axes"Between the Angles are"Equal"When thepositive Two measurement
When the projection surface is"Three axes"Between the Angles are"Not Equal"When thepositive three test

The positive axis of the space object is measured by the V-plane (xoz plane) as the projection plane, and the object is rotated to the y angle of the z-axis first.

Then the x-axis to-α angle, and finally to the V-plane projection, the transformation matrix is: T positive =tz? Tx? Tv

"Three-View vs. Axis mapping"

2> Oblique projection

"Parallel Projection Features"
Parallel projection keeps the "relevant proportionsof the object" unchanged;
The "Exact view" of each surface of an object is projected by a parallel projection;
no representation of the authenticity of the appearance of the three-dimensional object is given.
" axial projection" is formed by the "parallel projection method ", "Theviewpoint is in Infinity".


"Summary of three-dimensional graphics transformation"


According to the specific function of T3D in the transformation, the T3D can be divided into four matrices, namely:

Classification of plane geometry projections:

Three-dimensional image projection transformation--parallel projection