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

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OP 2 = OB2 + BP 2

= OA2 + AB2( ) + BP 2

OP = 32 + 42( ) + 52 = 50 = 7.071...

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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

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OP = a = [ax,ay,az ]

Laws of Algebra (for the field of real numbers R)

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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)

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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)

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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)

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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)

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Distributive law - links addition and multiplication

a, b, c ∈ Ra(b + c) = ab + ac multiplication is distriubtion over addition

Laws of Vector Algebra

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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!