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Introduction to Homogeneous Transformations & Robot Kinematics
Jennifer Kay, Rowan University Computer Science Department
January 2005
1. Drawing 3 Dimensional Frames in 2 Dimensions
We will be working in 3D coordinates, and will label the axes
x
,
y
, and
z
. Figure 1 contains a sample 3D coor
dinate frame.
Because we are representing 3D coordinate frames with 2D drawings, we have to agree on what these drawings
mean. Clearly the y axis in Figure 1 points to the right, and the z axis points up, but we have to come up with a
convention for what direction the x axis is pointing. Since the three axes must be perpendicular to each other, we
know that the x axis either points into the paper, or out of the paper. Most people instantly assume one or the
other is the case. To be able to view both cases, it helps to look at the axes overlaid on a cube. Consider the two
views of the same cube in Figure 2. In view (a) we are looking at the cube from below, in view (b) we are looking
at the cube from above. Let’s try and overlay the 3D coordinate frame from Figure 1 onto these two views.
Before you turn the page, make sure you can see both views of the cube in Figure 2!
x
y
z
Figure 1. A 3D coordinate frame.
Figure 2. Two views of the same
cube. The cube is missing the front
side, has a solid back and sides, and
patterned top and bottom. In view a
we are looking at the cube from
below, in view (b) we are looking at
the cube from above.
(a)
(b)
Copyright (C) 2003, 2005, Jennifer Kay. Permission is granted to make individual copies of this for educational
purposes as long as this copyright notice remains intact. Permission to make multiple copies of this for educational
purposes will generally be granted when an email request is sent to [email protected] (so I know who is using it). In
such cases, I will require that it is distributed for free (or for the actual cost of printing / duplication) and this copy
right notice remains intact.
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Figure 3 and Figure 4 show the same two views of the cube, this time with the 3D coordinate frame from Figure
1 overlaid onto the cube. Note that in Figure 3 the x axis points into the paper, away from you, and in Figure 4
the x axis is pointing out of the paper towards you!
For the purposes of this document, we will assume that Figure 4 shows the interpretation we will use. In other
words, if you see 3 axes drawn as they are in Figure 1, you should assume that the x axis points out of the paper
towards you. If you actually wanted the x axis to be pointing into the paper, you should use the illustration shown
in Figure 5.
Figure 3. A cube viewed from below.
The edge
labelled x points into the page away from you
.
x
y
z
Figure 4. The same cube viewed from above.
The edge labelled x points out of the page
towards you
.
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 Spring '10
 jons
 Cartesian Coordinate System, Mathematica, Euclidean geometry, Polar coordinate system, Rotx

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