Lecture 4 : More on Matrix Multiplication : Homogeneous Coordinates 2 -1 -1 4 0 Imagine the two matrices A = 2 -1 and B = . The 2 3 6 1 4 products AB and BA can be formed. As A has 3 rows and 2 columns while B has 2 rows and 3 columns, AB will have 3 rows
Lecture 10 (2010) : Transformations : Reection : Projection
into a plane
In two dimensions or in three dimensions, a point (or a vector) can be
reected in a line (two dimensions) or a plane (three dimensions). Once
again, the reection is carried out by me
Lecture 7 : Intersections of lines and planes : Ane Transformations
Intersections of 2 lines
Imagine the two lines with equations r1 = a + b and r2 = c + d.
These lines will intersect if values of and can be found so that r1 = r2.
There are several possib
Lecture 8 : Transformations : Translation : Sacling
Translation
A translation is a transformation that moves each point by a xed distance in a given direction.
In three dimensions, a translation where every point is moved by a distance
of x in the xdirect
Lecture 9 (2010) : Transformations : Rotation
In two dimensions, a rotation takes place around a particular point. This
point is left unchanged by the rotation and the coordinates of all other
points are changed.
The rotation takes place, anti-clockwise t
Lecture 6 : 2009 : Vectors, lines and planes
The vector equation of a line
A line can be described in terms of two vectors i.e. 1. The position vector of a point (any point) on the line 2. A vector parallel to the line A line which passes through the poin
Lecture 5 : Matrices and Linear Equations A system of equations of the form x + 2y = 7 2x y = 4 or 5p 6q + r = 4 2p + 3q 5r = 7 6p q + 4r = 2 is said to be a linear system. Such systems often occur in computer graphics and other areas of Computer Science.
Lecture 2 : The scalar product of two vectors.
The Scalar Product
Given two vectors, a and b, it is possible to form the scalar
product a b.
Before proceeding further, it is important to note that the
scalar produce is a scalar i.e. a number.
The scalar p
Lecture 1 : Introduction to Vectors
Right-Angle Trigonometry
Any shape or image can be broken down into smaller shapes.
Important within this is the role of right-angled triangles.
There are relationships between the sides and angles of rightangled triang
Lecture 3 : Introduction to Matrices A matrix is a rectangular array 1 2 2 5 1 6 3 or or 3 3 8 2 1 9 4 of numbers (or other entries) .
A matrix has clearly dened numbers of rows and columns 2 5 has 2 rows and 2 columns 3 8 1 6 3 has 2 rows and three colu
Lecture 11 : Transformations : Shearing : Combination of
Transformations
Shear
The Shear is a transformation where ONE coordinate is transformed by
an amount dependent on one of the remaining coordinates.
Basic Shape
A shear where y is varied by an amount