Mathematics for Graphics

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Mathematics for Graphics
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Objectives

• Introduce the elements of geometry
• Scalars
• Vectors
• Points
• Develop mathematical operations among them in a
  coordinate-free manner
• Define basic primitives
• Line segments
• Polygons

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3D Cartesian co-ordinates
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Basic Elements

• Geometry is the study of the relationships among
  objects in an n-dimensional space
• In computer graphics, we are interested in
  objects that exist in three dimensions
• Want a minimum set of primitives from which we
  can build more sophisticated objects
• We will need three basic elements
• Scalars
• Vectors
• Points

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Scalars

• Need three basic elements in geometry
• Scalars, Vectors, Points
• Scalars can be defined as members of sets which
  can be combined by two operations (addition and
  multiplication) obeying some fundamental axioms
  (associativity, commutivity, inverses)
• Examples include the real and complex number
  systems under the ordinary rules with which we
  are familiar
• Scalars alone have no geometric properties

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Vectors And Point

• We commonly use vectors to represent
• Points in space (i.e., location)
• Displacements from point to point
• Direction (i.e., orientation)
• But we want points and directions to behave
  differently
• Ex To translate something means to move it
  without changing its orientation
• Translation of a point different point
• Translation of a direction same direction

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Vectors

• Physical definition a vector is a quantity with
  two attributes
• Direction
• Magnitude
• Examples include
• Force
• Velocity
• Directed line segments
• Most important example for graphics
• Can map to other types

v
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Vector

• A quantity characterized by a magnitude and
  direction
• Can be represented by an arrow, where magnitude
  is the length of the arrow and the direction is
  given by slope of the line

v OP ??????? ?????? P-O
Y
P (2, 1)
1
v
X
2
O (0, 0)
A vector in 2D
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Vector Operations

• Every vector has an inverse
• Same magnitude but points in opposite direction
• Every vector can be multiplied by a scalar
• There is a zero vector
• Zero magnitude, undefined orientation
• The sum of any two vectors is a vector
• Use head-to-tail axiom

w
v
?v
v
-v
u
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Vectors Lack Position

• These vectors are identical
• Same direction and magnitude
• Vectors spaces insufficient for geometry
• Need points

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Vector Addition

• Addition of vectors follows the parallelogram law
  in 2D and the parallelepiped law in higher
  dimensions

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Vector Multiplication by a Scalar

• Multiplication by a scalar scales the vectors
  length appropriately (but does not affect
  direction)

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Subtraction
-v
u
v
-v
Inverse vector?
Can be seen as an addition of u (-1v)
-v
u
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Vector Magnitude
P2
v
P1

• The magnitude or norm of a vector of dimension
  3 is given by the standard Euclidean distance
  metric
• How about dimension n?

2D example
3D example
2
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Unit Vectors

• Vectors of length 1 are often termed unit vectors
  (a.k.a. normalised vectors).
• When we only wish to describe direction we use
  normalised vectors often to avoid redundancy
• For this and other reasons, we often need to
  normalise a vector
• e.g.

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Examples

• Suppose point P1(-1,-3) and P2(2,-7). Find
• Vector v obtained from these two points
• Norm of vector v
• Unit vector

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Dot Product

• Dot product (inner product) is defined as
• Note
• Therefore we can redefine magnitude in terms of
  the dot-product operator
• The dot product operator is commutative and
  associative.

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Dot Product

• The Dot Product can also be obtained from the
  following equation
• where q is the angle between the two vectors
• So, if we know the vectors u and v, then the dot
  product is useful for finding the angle between
  two vectors.
• Note that if we had already normalised the
  vectors u and v then it would simply be

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Dot Product Examples

• Find the angle between vectors 1, 1, 0 and
• 0, 1, 0?

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Additional Properties

• For any three vectors u, v, w and scalars a, b
• uv vu
• u(vw) uv uw
• (uv)w uw vw
• If uv 0 then u and v are orthogonal or
  perpendicular, where u and v are not zero vector

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Cross Product

• Used for defining orientation and constructing
  co-ordinate axes.
• Cross product defined as
• The result is a vector, perpendicular to the
  plane defined by u and v
• Magnitude

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Cross Product
Right Handed Coordinate System
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Cross Product

• Cross product is anti-commutative
• It is not associative
• Direction of resulting vector defined by operand
  order

R.H.S.
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Cross Product

• Consider, two vectors u , v ,
  the cross product u x v

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Vector Spaces

• A linear combination of vectors results in a new
  vector
• v a1v1 a2v2 anvn
• where a is any scalar
• If the only set of scalars such that
• a1v1 a2v2 anvn 0
• is a1 a2 a3 0
• then we say the vectors are linearly
  independent
• The dimension of a space is the greatest number
  of linearly independent vectors possible in a
  vector set
• For a vector space of dimension n, any set of n
  linearly independent vectors form a basis

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Coordinate Systems

• 3D vector space
• scalar component ?1, ?2, ?3
• basis vector v1, v2, v3
• define a coordinate system
• the origin fixed reference point
• column matrix

v2
origin
v1
v3
Coordinate System
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Points

• Location in space
• Operations allowed between points and vectors
• Point-point subtraction yields a vector
• Equivalent to point-vector addition

v P - Q
P v Q
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Affine Spaces

• Point a vector space
• Operations
• Vector-vector addition
• Scalar-vector multiplication
• Point-vector addition
• Scalar-scalar operations
• For any point define
• 1 P P
• 0 P 0 (zero vector)

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Lines (in Affine Space)

• Consider all points of the form
• P(a)P0 a d
• Set of all points that pass through P0 in the
  direction of the vector d
• Affine addition

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Parametric Form

• This form is known as the parametric form of the
  line
• More robust and general than other forms
• Extends to curves and surfaces
• Two-dimensional forms
• Explicit y mx h
• Implicit ax by c 0
• Parametric
• x(a) ax0 (1-a)x1
• y(a) ay0 (1-a)y1

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Rays and Line Segments

• If a gt 0, then P(a) is the ray leaving P0 in
  the direction d
• If we use two points to define v, then
• P( a) Qav Q a (R-Q) aR (1-a)Q
• For 0ltalt1 we get all the
• points on the line segment
• joining R and Q

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Convexity

• An object is convex iff for any two points in
  the object all points on the line segment between
  these points are also in the object

P
P
Q
Q
not convex
convex
concave
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Affine Sums

• Consider the sum
• Pa1P1a2P2..anPn
• Can show by induction that this sum makes sense
  iff
• a1a2..an1
• in which case we have the affine sum of the
  points P1,P2,..Pn
• If, in addition, aigt0, we have the convex hull
  of P1,P2,..Pn

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Convex Hull

• Smallest convex object containing P1,P2,..Pn
• Formed by shrink wrapping points

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Curves and Surfaces

• Curves are one parameter entities of the form
  P(a) where the function is nonlinear
• Surfaces are formed from two-parameter functions
  P(a, b)
• Linear functions give planes and polygons

P(a)
P(a, b)
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Planes

• A plane be determined by a point and two vectors
  or by three points

P
R
P(a,b)Ra(Q-R)b(P-Q)
P(a,b)Raubv
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Triangles
convex sum of S(a) and R

• convex sum of P and Q

for 0lta,blt1, we get all points in triangle
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Planes
R

• Extension of line

P
Q
n
v
P
normal to the plane
P0
u