Three-Dimensional Graphics

We will use a right-handed coordinate system.

(consistent with math)

Left-handed suitable to screens.

To transform from right to left, negate the z values.

Right Handed Space Left Handed Space

Homogeneous representation of 3D point:

|x|

|y|

|z|

|1|

(w=1 for a 3D point)

Transformations will be represented by 4x4 matrices.

Scaling:

| sx 0 0 0 |

| 0 sy 0 0 |

| 0 0 sz 0 |

| 0 0 0 1 |

Translation:

| 1 0 0 tx |

| 0 1 0 ty |

| 0 0 1 tz |

| 0 0 0 1 |

Shear (xy):

| 1 0 sh_x 0 |

| 0 1 sh_y 0 |

| 0 0 1 0 |

| 0 0 0 1 |

Rotation (along Z axis):

| cosq -sinq 0 0 |

| sinq cosq 0 0 |

| 0 0 1 0 |

| 0 0 0 1 |

Rotation (along X axis):

| 1 0 0 0 |

| 0 cosq -sinq 0 |

| 0 sinq cosq 0 |

| 0 0 0 1 |

Rotation (along Y axis):

(sign of sine reversed to maintain right-handed coordinate system)

| cosq 0 sinq 0 |

| 0 1 0 0 |

| -sinq 0 cosq 0 |

| 0 0 0 1 |

Compositing 3D transformations

We will cover 2 examples:

Example from text
Rotation about an arbitrary line in space.

The example from the text: Objective: Transform the directed line segments P₁P₂and P₁P₃from their starting position to their final position as indicated in the figure below. Thus,

Point P₁is to be translated to the origin,
P₁P₂is to lie on the positive Z axis and
P₁P₃is to lie in the positive y-axis half of the (y,z) plane.

Example 1: Composite 3D Rotation/Translation

4 Steps:

1) Translate P₁ to the origin
2) Rotate about the Y axis

3) Rotate about the X axis

4) Rotate about the Z axis

Rotation About an Arbitrary Axis in Space

Assume we want to perform a rotation about an axis in space passing through the point (x0, y0, z0) with direction cosines (cx, cy, cz) by d degrees.

How do we do this?

1) First of all, we want to translate by -(x0, y0, z0)= |T|.

2) Next, we rotate the axis into one of the principle axes, let's pick Z (|Rx|, |Ry|).

3) We rotate next by d degrees in Z ( |R_z(d)|).

4) Then we undo the rotations to align the axis.

5) We undo the translation: translate by (x0, y0, z0)

The tricky part is (2) above.

This is going to take 2 rotations,

1 about x (to place the axis in the xz plane) and

1 about y (to place the result coincident with the z axis).

Rotation by Alpha about x:

How do we determine Alpha?

Project the vector into the yz plane as shown below.

The y and z components are cy and cz, the direction cosines of the unit vector along the arbitrary axis.

It can be seen from the diagram below that

d= sqrt(cy2 + cz2), therefore

cos(a) = cz/d

sin(a) = cy/d

Rotation by Beta about y:

How do we determine Beta?

Similar to above:

determine the angle Beta to rotate the result into the Z axis:

The x component is cx and the z component is d.
It can be seen from the diagram that:

cos(Beta)= d = d/(length of the unit vector)

sin(Beta)= cx = cx/(length of the unit vector).

Final Transformation:

M = |T|-1 |Rx|-1 |Ry|-1 |Rd| |Ry| |Rx| |T|

If you are given 2 points instead, you can calculate the direction cosines as follows:

V =|(x1 -x0)|

|(y1 -y0)|

|(z1 -z0)|

cx = (x1 -x0)/ |V|
cy = (y1 -y0)/ |V|
cz = (z1 -z0)/ |V|, where |V| is the length of V

Transformations as a Change in Coordinate Systems

P⁽ⁱ⁾ is the representation of the point P in coordinate system i.

M_i<-jis the transformation that converts the representation of point in coordinate system j into coordinate system i.

P⁽ⁱ⁾ = M_i<-j* M_j<-k * P^(k)

The example Above:

M_1<-2 = T(4,2)

M_2<-3= T(2,3)* S(0.5, 0.5)

M_3<-4= T(6.7,1.8)* R(-45)

M_1<-4 = T(6,5)* S(0.5, 0.5) *T(6.7,1.8)*R(-45)

Note that M_i<-j = M_j<-i ^-1

The 3D Viewing Pipeline

Objects are modeled in object (modeling) space.

Transformations are applied to the objects to position them in world space.

View parameters are specified to define the view volume of the world, a projection plane, and the viewport on the screen.

Objects are clipped to this View volume.

The results are projected onto the projection plane (window) and finally mapped into the viewport.

Hidden objects are then removed, the objects scan converted and shaded if necessary.

Spaces

Object Space

definition of objects. Also called Modeling space.

World Space

where the scene and viewing specification is made

Eyespace (Normalized Viewing Space)

where eye point (COP) is at the origin looking down the Z axis.

3D Image Space

A 3D Perspected space.

Dimensions: -1:1 in x & y, 0:1 in Z.

Where Image space hidden surface algorithms work.

Screen Space (2D)

Coordinates 0:width, 0:height

The Computer Graphics Pipeline Viewing Process

Projections

We will look at several planar geometric 3D to 2D projection:

-Parallel Projections

Orthographic

Oblique

-Perspective

Projection of a 3D object is defined by

straight projection rays (projectors) emanating from the center of projection (COP)

passing through each point of the object and intersecting the projection plane.

Perspective Projections:

Distance from COP to projection plane is finite.

The projectors are not parallel & we specify a center of projection.

Center of Projection is also called the Perspective Reference Point

COP = PRP

Perspective foreshortening: the size of the perspective projection of the object varies inversely with the distance of the object from the center of projection.

Vanishing Point: The persepctive projections of any set of parallel lines that are not parallel to the projection plane converge to a vanishing point.

Example:

Example: 2 Point Perspective Projection

Parallel Projection

Distance from COP to projection plane is infinite.

Therefore, the projectors are parallel lines & we specify a direction of projection (DOP)

Orthographic: the direction of projection and the normal to the projection plane are the same.

(direction of projection is normal to the projection plane)

Example Orthographic Projection:

Axometric orthographic projections use planes of projection that are not normal to a principal axis.

(they therefore show mutiple face os an object.)

Isometric projection: projection plane normal makes equal angles with each principle axis

All 3 axis are equally foreshortened allowing measurements along the axes to be made with the same scale.

Example Isometric Projection:

Oblique projections : projection plane normal and the direction of projection differ.

Plane of projection is normal to a principle axis

Projectors are not normal to the projection plane

Example Oblique Projection

Classification of Geometric Projections:

Note: Most of the figures in this chapter are scanned from and copyrighted in Introduction to Computer Graphics by Foley, Van Dam, Feiner, Hughes, and Phillips, Addison Wesley 1994.