Date post: | 18-Jan-2018 |
Category: |
Documents |
Upload: | paulina-mckinney |
View: | 224 times |
Download: | 0 times |
Advanced Computer GraphicsCS32310
October 2012H Holstein
Coordinate Systems
• Mapping of points in space to tuple numbers• Existence of inverse mapping• René Descartes 1596-1650• 3D space
• 3 mutually perpendicular axes: x,y,z• Right handed convention• User defined position of origin and axis orientation
Distance from the origin O
A
B
€
OP 2 = OB2 + BP 2
= OA2 + AB2( ) + BP 2
OP = 32 + 42( ) + 52 = 50 = 7.071...
€
x 2 + y 2 + z2
Vectors (3D)The displacement of a point P (ax, ay, az) from the
origin O defines a vector a = [ax, ay, az]
Ordered 3-tuple.Magnitude and direction, but location unspecified.
A
B
a
€
OP = a = [ax,ay,az ]
Laws of Algebra (for the field of real numbers R)
€
Addition
a, b, c ∈ Ra + b = b + a commutative rulea + (b + c) = (a + b) + c associative rule a + 0 = a there exists an additive identity
Laws of Algebra (for the field of real numbers R)
€
Multiplication (operator often omitted)
a, b, c ∈ Rab = ba commutative rulea(bc) = (ab)c associative rule a1 = a there exists an multiplicative identity
Laws of Algebra (for the field of real numbers R)
€
Subtraction is defined in terms of addition of an inverse
a, b ∈ Ra + a = 0 additive inverse, also written as (−a)
a + b ≡ a − b definition of substraction
Laws of Algebra (for the field of real numbers R)
€
Division is defined in terms of multiplication by an inverse
a, b ∈ R
a a−1 =1 multiplicative inverse, provided a ≠ 0
ab−1 ≡ a /b definition of subtraction
Laws of Algebra (for the field of real numbers R)
€
Distributive law - links addition and multiplication
a, b, c ∈ Ra(b + c) = ab + ac multiplication is distriubtion over addition
Laws of Vector Algebra
€
Distributive law - links addition and multiplication
a, b, c ∈ Ra(b + c) = ab + ac multiplication is distriubtion over addition
L E A R N !!
L E A R N !!
x y z rule!