---
title: "03-2d-transforms"
tags: 
- lecture
- cosc342
---

look into how colours work together

Points lines
- point is 2d location $(u,v)$
- two points define a line
- a polyline with k segments is a sequence of k+1 points
- a polygon is a polyline where the beginning and ened are the same, we often omit the duplicate point
- points are vectors

> [!INFO] polygon and polylines will be specifies in the code

> [!INFO] $[u v]^T$ high T indicates vector 

coordingate systems
- mathematical
- image based
- matrix based
- ![|300](https://i.imgur.com/m6OAA5T.png)

> [!INFO] make sure to check you are using the right coordinate system

transformations
- translation
> [!INFO]  value of change (delta) for each coordinate. for a shape, apply the transformation to each point
- scaling
- rotation
	- rotate by an abgle about the origin
	- rotation from U towards V, not anti/clockwise
	- $[u',v'] = [cos(0) - sin(0), sin(0) cos(0)][u,v]$
- inverse
	- inverse of translate is translating by negative
	- inverse of scaling by $s$ is scaling by $\frac{1}{s}$
	- inverse of rotating by $\theta$ is rotating by $-\theta$
	- inverse of rotation matrix is its transpose
		- ![](https://i.imgur.com/HSiqyQb.png)

- combinations
	- e.g., rotate 45 about (2,1)
		-  shift by (-2,-1)
		-  rotate by 45
		-  shift by (2,1)
		-  ![](https://i.imgur.com/NI6luaG.png)

homogenous coordinates
- represent 2D points as 3 points
- all linear transformations become 3x3 matrices
- ![](https://i.imgur.com/aksrjQw.png)

>[!DIFFICULT] 

homogenous transforms