Vector Spaces and Basis
Transforms
Co-ordinates
Bases
Using a matrix to represent a basis
World space, camera space, object space
Building interactive objects in the virtual world
Docsity.com
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Understanding Coordinates and Basis Vectors in 3D Spaces, Slides of Fundamentals of E-Commerce
The concept of coordinates and basis vectors in 3d spaces. It discusses how coordinates define the location of a point in a space using a set of orthogonal unit vectors called a basis. The document also covers the importance of having a complete basis to describe any point in space and demonstrates how to find the coordinates of a point relative to a different basis. Additionally, it introduces the concept of using matrices to represent and transform points between different bases.
Typology: Slides
2012/2013
1 / 24
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Understanding Coordinates and Basis Vectors in 3D Spaces and more Slides Fundamentals of E-Commerce in PDF only on Docsity!
Vector Spaces and BasisTransforms
Co-ordinates
Bases
Using a matrix to represent a basis
World space, camera space, object space
Building interactive objects in the virtual world
What are co-ordinates?
Well, yes... in a way. But even if they’re the same point,you might describe them using different co-ordinates.
Co-ordinate spaces
Consider the point [ x, y]:
Is this the same point?
What is a basis?
Basis Vectors
If P = [ x, y, z ] then
P = [ 1, 0, 0 ] * x +
// X axis
[ 0, 1, 0 ] * y +
// Y axis
[ 0, 0, 1 ] * z
// Z axis
These three vectors are all that’s needed to uniquely definethe location of any point in the world. Any point can bedescribed as a sum of one or more of these three vectors,multiplied by some scalar coefficient.
These three vectors, X axis, Y axis, and Z axis, form what’scalled a
basis
, or
co-ordinate system
The particular vectors shown here--
[ 1, 0, 0 ], [ 0, 1, 0 ], and [ 0, 0, 1 ]
--form what’s called the
identity basis
What is a basis?
A basis is a set of n orthogonal unit vectors
where each vector is of dimension n.
The term
orthogonal
means
perpendicular
A basis forms a co-ordinate system which can be used tospecify the location of any point in space as a sum of one ormore of the basis vectors multiplied by scalar coefficients.
Examples:
[ 1, 0, 0 ], [ 0, 1, 0 ], and [ 0, 0, 1 ] is a basis.
[ 1, 0, 0 ], [ 0, 1, 0 ], and [ 0, 0, 2 ] is not; Z is not length one.
[ 1, 0, 0 ], [ 0, 1, 0 ], and [ 0, 1, 1 ] is not; Z is not perpendicular to Y.
Building a new basis
Remember the rotation matrix around the Y axis?
[
cos(45º)
0
-sin(45º)
]
R
=
[
0
1
0
]
[
sin(45º)
0
cos(45º)
]
which can be simplified, because
cos(45º)
=
sin(45º)
=
2/
as:
[
2/
0
2/
]
R
=
[
0
1
0
]
[
2/
0
2/
]
We can multiply X, Y, and Z by R:
X’ = R * [1,0,0] =
[
2/2 ]
Y’ = R * [0,1,0] =
[
0 ]
Z’ = R * [0,0,1] =
[ -
2/2 ]
Building a new basis
Now we can build the basis B:
X
[
2/2 ]
Y
[
0 ]
Z
[
2/2 ]
…which is just like our usual identity basis except that the X and
Z vectors are rotated 45 degrees in the X Z plane.
Is this a basis?
All three vectors are unit vectors
2
2
2
All three vectors are orthogonal
X • Y = (
[
2/2]
) • (
[0,1,0]
) = 0+0+0 = 0
X • Z = 0Y • Z = 0
There are three vectors in three dimensions... It’s a basis!
Using different bases to describe the samepoint in space
Finding the co-ordinates (a,b,c) :
We know that
(
2/2,
0,
2/2)
a
(1,0,0)
0
(
0,
1,
b
=
(0,1,0)
0
(-
2/2,
0,
2/2)
c
(0,0,1)
z
Multiply through on both sides:
(
(
2/2)a,
0,
(
2/2)a)
(1,0,0)
(
0,
b,
=
(0,1,0)
(-(
2/2)c,
0,
(
2/2)c)
(0,0,z)
Add through on both sides, x’es with x’es, y’s with y’s:
((
2/2)a-(
2/2)c,
b,
(
2/2)a+(
2/2)c)
=
(0,0,z)
Using different bases to describe the samepoint in space
From ((
2/2)a-(
2/2)c,
b,
(
2/2)a+(
2/2)c)
=
(0,0,z)
we know that
(
2/2)a-(
2/2)c
=
0
(
2/2)a+(
2/2)c
=
z
b
=
0
and so
a-c
=
0
a+c
=
2
z
/
2
b
=
0
and thus
a
=
z
/
2
b
=
0
c
=
z
/
2
...so the co-ordinates of P in basis B are (z/
2, 0, z/
Using matrices to describe bases
To transform a point from one basis to another basis, wecan build a matrix representation of the basis:
[
[ 1 ]
[ 0 ]
[ 0 ]
]
I = [
[ 0 ]
[ 1 ]
[ 0 ]
]
[
[ 0 ]
[ 0 ]
[ 1 ]
]
and
[
[ (
2/2) ]
[ 0 ]
[ (
2/2)
]
]
B = [
[
0
]
[ 1 ]
[
0
]
]
[
[-(
2/2) ]
[ 0 ]
[ (
2/2)
]
]
or in general, the matrix representation of a basis is the matrix
whose columns are the vectors of the basis:
[
[
]
[
]
[
]
]
M = [
[ X ]
[ Y ]
[ Z ]
]
[
[
]
[
]
[
]
]
Using matrices to describe bases
What can this representation do?
Try a simple matrix multiply:
P’ = B * P
I
or rather
[
[ (
2/2) ]
[ 0 ]
[ (
2/2)
]
]
[ 0 ]
P’ = [
[
0
]
[ 1 ]
[
0
]
] * [ 0 ]
[
[-(
2/2) ]
[ 0 ]
[ (
2/2)
]
]
[ z ]
which means
P’ = [ (
2/2)z, 0, (
2/2)z ]
which is the co-ordinates for P
B
World Space and Camera Space
The three vectors defining a camera,
lookFrom,
lookAt
lookUp
…yield three perpendicular vectors X, Y, and Z:
Z = (lookAt – lookFrom).normalised()
Y = lookUp.normalised()
X = Y
Z
These three vectors define a basis B
C
, in which
the camera looks up the Z axis towards (0, 0, 1).
[
[
]
[
]
[
]
]
M
C
=
[
[ X ]
[ Y ]
[ Z ]
]
[
[
]
[
]
[
]
]
World Space and Camera Space
“Camera space” vs “World space” :
Given any point P
I
in the virtual world, if you subtract the
lookFrom of the camera from P
I
and then multiply the
resulting vector by M
C
, you will have the
position of P relative
to the camera
P
C
= M
C
- (P
I
- lookFrom)
By the same token, if you take a point in camera space—say,the point on the screen where the user just clicked (if I click atx=20, y=20 then in camera space that’s the point (20, 20, 0))then that point can be converted to world space by
P
I
= (M
C
- P
C
) + lookFrom
In graphics we generally refer to P
I
as being in
world space
and P
C
as being in
camera space
Simplifying world and camera space
The camera-to-world equations can be rewrittenas a single matrix transformation.
P
C
= M
C
* (P
I
- lookFrom)
P
I
= (M
C
* P
C
) + lookFrom
The operation (P
I
- lookFrom) is a translation operation,
shifting P by lookFrom units. Any translation can beexpressed in matrix form:
[
1
0
0
–lookFrom.x
]
T
lookFrom
= [
0
1
0
–lookFrom.y
]
[
0
0
1
–lookFrom.z
]
[
0
0
0
1
]
Multiplying any point P by the matrix
T
lookFrom
, above, would
subtract lookFrom from P.
Simplifying world and camera space
We can extend the matrix M
C
, the matrix which encapsulates
the transformation from world space to camera space, into a4x4 matrix.
[
0 ]
M
C4x
= [
M
C
0 ]
[
0 ]
[ 0 0 0 1 ]
Concatenating M
C4x
onto
T
lookFrom
creates a matrix which will
first translate any point it multiplies, then transform that pointinto a new basis:
WorldToCamera
= M
C
4x
T
lookFrom
which allows us to write
P
C
WorldToCamera
* P
I
P
I
WorldToCamera
-
* P
C