Computational Geometry

Vectors, Translation, Rotation

George Georgiev

http://academy.telerik.com

Technical Trainer

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George AtanasovFront-End Developer

Table of Contents Analytic geometry

Vectors and Points

Vector math Addition / Translation

Subtraction

Multiplication / Scaling

2D Rotation

Vector length

Programming it2

Analytic geometryThe math

Analytic geometry

Also called Cartesian geometry Coordinate system

Applies algebraic principles to geometry Algebra

Analysis

4

Analytic geometry Points (vertices)

Represented by coordinates 2D – (X, Y);

3D – (X, Y, Z);

Etc..

Points are locations inspace

5

Analytic geometry Points

Example – A (3, 2); H (-1.5, 3);

6

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)

7

Analytic geometry

8

A Point A vector to the point

Vector mathIt's easy

Vector math Addition

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)

10

Vector math Multiplication

By a number 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)11

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’)12

Vector math 2D Rotation

Positive angle, counter-clockwise rotation

Negative angle, clockwise rotation

13

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

14

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 length15

Vector mathLive Demo

Programming geometry

Point classes, functions, operators

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

18

Programming geometryLive Demo

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