advanced computer graphics cs32310 october 2012 h holstein
TRANSCRIPT
![Page 1: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/1.jpg)
Advanced Computer GraphicsCS32310
October 2012H Holstein
![Page 2: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/2.jpg)
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
![Page 3: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/3.jpg)
![Page 4: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/4.jpg)
![Page 5: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/5.jpg)
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
![Page 6: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/6.jpg)
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 ]
![Page 7: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/7.jpg)
![Page 8: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/8.jpg)
![Page 9: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/9.jpg)
![Page 10: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/10.jpg)
![Page 11: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/11.jpg)
![Page 12: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/12.jpg)
![Page 13: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/13.jpg)
![Page 14: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/14.jpg)
![Page 15: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/15.jpg)
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
![Page 16: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/16.jpg)
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
![Page 17: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/17.jpg)
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
![Page 18: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/18.jpg)
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
![Page 19: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/19.jpg)
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
![Page 20: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/20.jpg)
Laws of Vector Algebra
€
Distributive law - links addition and multiplication
a, b, c ∈ Ra(b + c) = ab + ac multiplication is distriubtion over addition
![Page 21: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/21.jpg)
![Page 22: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/22.jpg)
L E A R N !!
![Page 23: Advanced Computer Graphics CS32310 October 2012 H Holstein](https://reader035.vdocuments.us/reader035/viewer/2022081507/5a4d1b7b7f8b9ab0599b9333/html5/thumbnails/23.jpg)
L E A R N !!
x y z rule!