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Notation, Point & Vector Basics Introduction To Computer Graphics Arizona State University

This document introduces some basic concepts in 2D and 3D computer graphics notation. It defines a 2D coordinate system with an origin o and basis vectors e1 and e2. Points are represented by their coordinates relative to this system. Vectors describe displacements with direction and magnitude. The sum of a point and a vector yields another point. Vector addition follows the parallelogram rule. Point addition uses barycentric coordinates to interpolate between points and yield a point on the line between them. Certain rules allow point operations to be coordinate system independent.

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83 views5 pages

Notation, Point & Vector Basics Introduction To Computer Graphics Arizona State University

This document introduces some basic concepts in 2D and 3D computer graphics notation. It defines a 2D coordinate system with an origin o and basis vectors e1 and e2. Points are represented by their coordinates relative to this system. Vectors describe displacements with direction and magnitude. The sum of a point and a vector yields another point. Vector addition follows the parallelogram rule. Point addition uses barycentric coordinates to interpolate between points and yield a point on the line between them. Certain rules allow point operations to be coordinate system independent.

Uploaded by

Lyn Lyn Villamil
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Notation, Point & Vector Basics

Introduction to Computer Graphics


Arizona State University

Dianne Hansford

August 22, 2007

1 Notation

Lets focus on 2D. After that, 3D is easy!

The [e1 , e2 ]-coordinate system is defined by the origin o, and the e1 and e2
vectors:
0 1 0
o= , e1 = , e2 = .
0 0 1

A point p describes a location and its coordinates are defined as



p
p= 1 .
p2

Therefore, a points location is defined by its coordinates in this coordinate


system: p = o + p1 e1 + p2 e2 . Sometimes it is convenient to label the
coordintes of p as px and py .

A vector v describes a displacement (direction and distance) and its com-


ponents take the form
v
v= 1 ,
v2
and thus v = v1 e1 + v2 e2 . Figure 1 illustrates this basic geometry.

1
p
e2

0 e1

Figure 1: Point and vector notation.

2 Point and Vector Combinations

As illustrated in Figure 2, the sum of a point and a vector is a point:

q = p + v. (1)

Vector addition may be described in terms of the parallelogram rule, as


illustrated in Figure 3. Therefore, any scalar values s and t are acceptable
in forming u = sv + tw.

To formulate point addition, lets revisit (1) and the idea that a point plus
a vector results in a point. As in Figure 4, we may form a point on the line
through p and q as

r = p + tv (2)
= p + t(q p) (3)
= (1 t)p + tq, (4)

therefore, the coefficients for point addition must sum to one. Such a re-
quirement on the coefficients is called a barycentric combination. Another
name for (4) is linear interpolation. Notice that when the parameter t = 0,
r = p, when t = 1, r = q, and when t = 1/2, r is the midpoint between p
and q.

2
r
v

Figure 2: The sum of a point and vector yields a point.

Thus we see that the relative positioning of p, q, r may be characterized by


the parameter
||r p||
t= ,
||q p||
using a signed distance, as illustrated in Figure 5. The relationship between
the three points is also reflected in the quotient
t ||r p||
ratio(p, r, q) = = ,
1t ||q r||
and this ratio is also illustrated in Figure 5.

Point addition comes with some rules for two reasons.

1. We would like coordinate-system independent operations. For exam-


ple, as illustrated in Figure 6, averaging two points should always yield
the same point.
2. We want a method to construct a new point within the confines of the
geometry defined by the given points. For example, as illustrated in
Figure 4, we would like to construct a point on the line defined by p
and q.

For more information, examples, and exercises, see Practical Linear Al-
gebra A Geometry Toolbox by G. Farin & D. Hansford, A K
Peters, 2005.

3
w w-v v+w

v-w
v

Figure 3: The parallelogram rule encapsulates vector addition.

Figure 4: The operation r = (1 t)p + tq results in a point on the line


defined by p and q.

4
1:3
t=1/4 t=3/2

3:1
t=3/4 t= -1/10

t : 1-t
t=1/2
p r q

Figure 5: Examples of the ratio of three points and the corresponding pa-
rameter t.

r? r?
p

Figure 6: The result of the operation r = p + q is dependent on the coordi-


nate system.

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