1. Computational Geometry
Vectors, Translation, Rotation
George Georgiev
Telerik Corporation

2. Table of Contents Analytic geometry Vector math Programming it
Vectors and Points
Vector math
Addition / Translation
Subtraction
Multiplication / Scaling
2D Rotation
Vector length
Programming it

3. Analytic geometry
The math

4. Analytic geometry Also called Cartesian geometry
Coordinate system
Applies algebraic principles to geometry
Algebra
Analysis

5. Analytic geometry Points (vertices) Represented by coordinates
2D – (X, Y);
3D – (X, Y, Z);
Etc..
Points are locations in space

6. Analytic geometry
Points
Example – A (3, 2); H (-1.5, 3);

7. Analytic geometry Vectors Represented exactly the same way
Different only by concept
Have
Direction
Magnitude (length)
Computers understand vectors
Points – vectors added to the beginning of the coordinate system (0, 0)

8. Analytic geometry
A Point
A vector to the point

9. Vector math
It's easy

10. Vector math Addition Subtraction
Vectors can be added just like numbers
Called translation when applied to a point
Example
A(5, 10) + B(-3, 7) = C(2, 17)
Subtraction
Subtracting two points gives the vector from the second to the first
Example: A(5, 10) - B(-3, 7) = V(8, 3)

11. Vector math Multiplication By a number Called uniform scaling
Multiple each coordinate by the number
V(5, 7) * 3 = V’(15, 21)
Called uniform scaling
Division works the same way
You can always multiply by 1/3 (that’s 0.33)
V(15, 21) / 3 = V’(5, 7)
V(15, 21) * 0.33 = V’(5, 7)

12. Vector math 2D Rotation A little more complicated
Multiply two of the vector’s coordinates according to a formula
The formula
The vector V(x, y) rotated by the angle ALPHA
x’ = x*cos(ALPHA) - y*sin(ALPHA)
y’ = x*sin(ALPHA) + y*cos(ALPHA)
The resulting vector is V’(x’, y’)

13. Vector math 2D Rotation Positive angle, counter-clockwise rotation
Negative angle, clockwise rotation

14. Vector math Vector length Calculated with the Pythagorean theorem
The vector’s x and y components are perpendicular
The vector’s length is: length = sqrt (x*x + y*y)
Where sqrt is the square root

15. Vector math Unit vector A unit vector is a vector with a length of 1
A unit vector multiplied by a number has length equal to the number
‘Converting’ a vector to a unit vector
Called Normalization
Get the length of the vector
Divide the vector by it’s length

16. *
Vector math
Live Demo
(c) 2008 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

17. Point classes, functions, operators
Programming geometry
Point classes, functions, operators

18. Programming geometry We need A class describing a point/vector
Operators for vector addition, multiplication
A rotation method
A normalization method
A length calculation method

19. Programming geometry Live Demo *
(c) 2008 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

20. Computational geometry? ? ? ? ?
Questions? ? ? ? ? ? ?