eScience Lectures Notes : Transformation in 3D

Slide 1 : 1 / 19 : Three-Dimensional Transformations

Three-Dimensional Transformations


Slide 2 : 2 / 19 : Three-Dimensional Transformations

Three-Dimensional Transformations

The 3-D graphics pipeline

Rigid-body transforms

Homogeneous coordinates


Slide 3 : 3 / 19 : The 3-D Graphics Pipeline

The 3-D Graphics Pipeline


  • Almost every disscusion of 3-D graphics begins here

  • Seldom are any two versions drawn the same way

  • Seldom are any two versions implemented the same way

  • Primitives are processed in a set series of steps

  • Each step forwards its result on to the next step


Nearly all discussions of three-dimensional computer graphics start with the graphics pipeline. However, the notion of "the graphics pipeline" is somewhat of a misnomer. Seldom are any two depictions, or implementations for that matter, of a graphics pipeline identical.
The basic idea is that the processing of information in 3-dimensional computer graphics occurs in a series of steps, where each step generates results for the successive one.
The process starts with models that are usually described by three-dimensional coordinates (x, y, z). The models usually define either solids or the boundaries of object. Each of these modeling styles has a name. In solid modeling the primitives used to describe an object have volume. In boundary representations, or B-reps, the objects are defined in terms of their skins, or two-dimensional surfaces that approximate the interfaces between different material types. These models are usually defined within there own coordinate system, with their own origins and scales of measurement.
The first step in the rendering pipeline is to transform an object from it's own model space to a common coordinate space, called world space, in which all objects, light sources, and the viewer coexist. This step is called the modeling transformation stage.
The next step is an optimization. In the trivial rejection stage of the rendering pipeline we attempt to eliminate any objects that cannot possibly be seen. This removes some of the scene description from further processing by the rest of the pipeline.
Next we illuminate the objects that might possibly be seen giving them colors based on their material properties and the light sources in the scene. There are many different ways to accomplish this illumination depending the shading model and the surface model.
After illumination we perform another change in coordinate system that maps the viewing position to the origin, and the viewing plane to some desired position. This viewing transformation moves objects from world space into eye space.
Next we perform clipping of the scene's objects within a three dimensional viewing volume called a viewing frustum. This step totally eliminates any objects (and pieces of objects) that are not visible in the image.
In the next step we actually project the objects into two-dimensions. The transformations is from eye-space to screen-space.
In the rasterization step we scan-convert the object into pixels. This will often involve interpolating parameters as we go along.

But we already know how to do this (This what we've been talking about thus far in the semester).

To summarize, almost every step in the rendering pipeline involves a change of coordinate systems. This transformation process is central to understanding three-dimensional computer graphics.

and it will be our starting point for our venture into 3-D computer graphics. By the way, it all about math.... but I warned you!

Slide 4 : 4 / 19 : Modeling transformations

Modeling transformations

  • We starts with 3-D models defined in their own model space

  • Modeling transformations orient models within a common coordinate system called world space

  • All objects, light sources, and the viewer live in world space

  • Trivial rejection attempts to eliminate objects that cannot possibly be seen (an optimization)


Slide 5 : 5 / 19 : Illumination


  • Next we illuminate potentially visible objects

  • Object colors are determined by their material properties, and the light sources in the scene

  • Illumination algorithm depends on the shading model and the surface model


Slide 6 : 6 / 19 : Viewing Transformation

Viewing Transformation

  • Another change of coordinate systems

  • Maps points from world space into eye space

  • Viewing position is transformed to the origin

  • Viewing direction is oriented along some axis

  • A viewing volume is defined


Slide 7 : 7 / 19 : Clipping and Projection

Clipping and Projection

  • Next we perform clipping of the scene's objects againist a three dimensional viewing volume called a viewing frustum

  • This step totally eliminates any objects (and pieces of objects) that are not visible in the image

  • Next the objects are projected into two-dimensions

  • Transformation from eye space to screen space




View Frustum

view frustum is a semi-infinite pyramid, although the base is usually rectangular rather than square because the screen isn't square. In practice, however, it is normal to chop off the frustum near the point and somewhere further down towards the base. The view frustum defines what parts of the 3D world can be seen from the camera position when looking through a window (usually the screen). Here's a little piccie to help clarify things a bit :You can define a frustum using the view rectangle, the camera origin and the near- and far-plane distances from the camera origin. This information can then be used to make the top, bottom, left, right, near and far planes. All the planes have their normals facing into the frustum (by convention, doesn't matter so long as they either all point inwards or all point outwards).
Once you have these 6 planes it is easy to determine whether any given point is inside the frustum or not - you just make sure it is above every plane. You just test it against each plane in turn, rejecting the point if it fails at any point in the test.


Slide 8 : 8 / 19 : Rasterization and Display

Rasterization and Display


  • One last transformation from our screen-space coordinates into a viewport coordinates

  • The rasterization step scan converts the object into pixels

  • Involve interpolating parameters as we go

  • Purely 2D operation

Almost every step in the rendering pipeline involves a change of coordinate systems. Transformations are central to understanding three-dimensional computer graphics.

Slide 9 : 9 / 19 : Rigid-Body Transformations

Rigid-Body Transformations

First we begin with a representation for 3-D points. We will consider points as column vectors. Thus, a typical point with coordinates (x, y, z) is represented as:

or... as ...

This is not the only possible representation. You may encounter textbooks that consider points as row vectors. What is most important is that you use a consistent representation.



Slide 10 : 10 / 19 : Homogenous coordinate

Homogenous coordinate


Slide 11 : 11 / 19 : Translations


Objects are usually defined relative to their own coordinate system. We can translate points in space to new positions by adding offsets to their coordinates, as shown in the following vector equation.

The following figure shows the effect of translating a teapot.

Slide 12 : 12 / 19 : Rotations


Rotations in three-dimensions are considerably more complicated than two-dimensional rotations. In general, rotations are specified by a rotation axis and an angle. In two-dimensions there is only one choice of a rotation axis that leaves points in the plane.

Right hand thumb rule

If the thumb points towards the positif axis (in the same direction as the vector that define the rotation) then positif rotations follow the other finger of the right hand.

A special case : Positive rotation directions about the coordinate axes are counterclockwise, when looking toward the origin from a positive coordinate position on each axis

Coordinate-Axes Rotations

Around the x axis Around the y axis Around the z axis

How to get that these matrix ? let's have a look first at the rotation around the z axis


Slide 13 : 13 / 19 : General Three dimensional Rotations

General Three dimensional Rotations

A rotation matrix for any axis that does not coincide with a coordinate axis can be set up as a composite transformation involving combination of translations and the coordinate-axes rotations.

  1. Translate the object so that the rotation axis passes through the coordinate origin

  2. Rotate the object so that the axis rotation coincides with one of the coordinate axes

  3. Perform the specified rotation about that coordinate axis

  4. Apply inverse rotation axis back to its original orientation

  5. Apply the inverse translation to bring the rotation axis back to its original position


In Java 3D no way to define directly a general rotation, but at least, it is possible to easily define a rotation around an axis that goes trough the origin

Slide 14 : 14 / 19 : Properties of the Rotation Matrix

Properties of the Rotation Matrix

Or, what is so special about a special orthogonal?

In case you missed it, a rotation matrix is a special orthogonal matrix. By definition, a special orthogonal matrix has these properties:


Where AT is the transpose of A and I is the identity matrix, and

det A = 1.

This isn't really very helpful. A more helpful set of properties is provided by Michael E. Pique in Graphics Gems (Glassner, Academic Press, 1990):

  1. R is normalized: the squares of the elements in any row or column sum to 1.

  2. R is orthogonal: the dot product of any pair of rows or any pair of columns is 0.

  3. The rows of R represent the coordinates in the original space of unit vectors along the coordinate axes of the rotated space. (Figure 1).

  4. The columns of R represent the coordinates in the rotated space of unit vectors along the axes of the original space.

Properties 1 and 2 are useful for verifying that a matrix is a rotation matrix. If you manipulate a matrix, and you want to make sure that you still have a rotation matrix, sum the squares of any row or column. If the result is not 1, then you have surely done something wrong. If the result is 1, chances are you are on the right track.

Property 3 is useful for forward motion.

Slide 15 : 15 / 19 : Euclidean Transformations

Euclidean Transformations

For points we can combine the actions of rotation about an arbitrary axis followed by a translation into a single 6-parameter transformation.

This set of transformations forms the algebraic group of the euclidiean transformations.

Slide 16 : 16 / 19 : More Modeling Transforms

More Modeling Transforms

It is possible to expand our repertoire of modeling transformations. We can add Similitudes, Affines, and Projective groups of operators. However, most of these are either not that useful in everyday practice, or they are commonly reduced to one or two special cases, which generalize via a series of more typical operations.
Among those transforms remaining, scales and shears are the most common:

We can generate any 3-D affine transformation using a combination of a rotation, a scale, a shear, and a translation.

shear [n.]

(§ Homonym: sheer)
1. A machine that cuts sheet metal by passing a blade through it.
2. (Physics) A deformation of an object in which parallel planes remain parallel but are shifted in a direction parallel to themselves; "the shear changed the quadrilateral into a parallelogram."


Slide 17 : 17 / 19 : Other way to look at rotation

Other way to look at rotation

Roll, Pitch, Yaw

Imagine three lines running through an airplane and intersecting at right angles at the airplane's center of gravity.




Rotation around the front-to-back

Rotation around the side-to-side axis

Rotation around the vertical axis


Euler Angles

written by Ian Humphrey <>

The Euler Angles (a, b, g) comprise a parametrisation of arbitrary rotations in three-dimensional space.
The description in this document coincides with that of "Angular Momentum in Quantum Mechanics" by Edmonds and "Angular Momentum" by Brink and Satchler, two standard texts in the field.

The effect of Euler angle rotation R=(a, b, g) is briefly as follows:
1. We start in frame S with cartesian axes, x, y and z.
2. We begin with a positive (anti-clockwise) rotation of magnitude a about the z axis of S. The resulting frame is called S'.
3. Next a positive rotation of magnitude b about the y' axis of frame S'. The resulting frame is called S''.
4. Finally a positive rotation of magnitude g about the z'' axis of S''. The resulting frame is called S'''.

the phycist definition :

Rotation of a about z

Rotation of b about y'

Rotation of g about z''



Slide 18 : 18 / 19 : Some more reflexions

Some more thoughts

How to get a 3D matrix rotation around a coordinate axis

How to get a 3D matrix rotation around any axis

Difference between R.T and T.R

How to move around a central point

Why Pitch, Roll, Yaw is not the same than Yaw, Pitch, Roll...

... The Truth is Out There ...

rotation are not commutative (has anybody proved the contrary ?)

what will happen in a real plane ?

How to get the Pitch from a general Transformation ?

... that depend of the way you want to define the pitch...

Just do only incremental stuff

So, how to apply some Pitchy relative rotation whatever your initial point is ?


Slide 19 : 19 / 19 : The Metaphors

The Metaphors

World in the Hand

When move the viewpoint around the centre of your Local node, looking always towards the centre of the basis, and more precisely towards an object placed in that centre.

This way to move around the Viewpoint is usually known as the "world in the hand" Metaphor.


A Metaphor is a way to explain to users of a program how to use its interface. You say "Do it as if you where doing something else that you already know". The most well-known ones are of course the "Desktop" 2D Metaphor invented by Xerox, developed and popularised by Apple. It says that to access the information available in a computer, you should have the same behaviour as in front of a real desktop. You could open a "folder"  from which you get a "file" that you could "open", "write" into and "close"...

As 3D world is more difficult to work with than 2D one, there are lots of metaphors around to represent the way to deal with that sort of world.

The Flying metaphor

Flying Plane

   Can roll, pitch and yaw the view and can move forwards (along the -ve z' axis).

Flying Saucer

   Can move up, down, left, right, forward and backwards relative to local heading (ie along the x', y' and z' axes).

Flying Superman

Other Metaphors

On the Ground metaphor

The Walking metaphor
The Driving metaphor

The be yourself and move around metaphor

... the virtual reality metaphor (Pure schema)

What about interaction (required for the implementation of certain metaphors)?

Mouse Picking

A ray between the mouse and your eye

The magic wand

This is just was selection... what about manipulation of objects in the scene...


the object in a cylinder

the object in a sphere