Introduction

- Points & vectors are represented by 3 real numbers – they represent the
**signed distance**between the origin of a line and the position of a point on it e.g. a cross on a line. The cross is the origin. - Origin is the point of reference from which the distance to other points is measured.
- If the point is to the
**right > 0 (or positive)** - If the point is to the
**left is negative** - With our line and origin we add additional marks at regular intervals either side of the origin to turn it into a ruler
- The ruler is used to measure the
**coordinate**(signed distance) of a point from the origin - The ruler defines the axis
- Point isn’t on axis;
- (assuming ruler is horizontal) drop a line perpendicular to it that would hit the point
- distance from origin to intersection of the point with vertical line = coordinate
- That’s how to coordinate a point with respect to the axis

Dimensions and Cartesian Coordinate Systems

- Assume horizon ruler (mentioned above) is the x axis
- Draw a perpendicular line through the origin – y axis
- For any point, get coordinates by:
- Drawing perpendicular lines through each axis to the point
- Then draw line from origin to the intersection of the lines

- We can now find 2 coordinates one on the x and one on the y
- By have two axes we have created a 2D space called a plane
- If there are two axes perpendicular to each other then it is a
**Cartesian Coordinate System** **Ordered pair**commonly used to denote the coordinates of a point (x, y)

- To find coordinates of a point in a system when the coordinates are known in one, use formula from previous post
- Translation – adding values to a point to move it
- Scale – multiply coordinates by a value

Third Dimension

- Simple extension of 2D system
- Addition of depth; z axis
- Euclidean space – 3D coordinate systems
- Three axes = basis = set of independent linear vectors, combined can represent every vector or point in system
- Vectors in a set are linearly independent only if they can’t be written as a combination of the other vectors in that set
- Change of basis is the same as the change of the coordinate system

Left Handed vs Right Handed Coordinate Systems (L/RHCS)

- Left handed – right vector points left
- Right handed – right vector points right
- Introduced by John Ambros Fleming
- Relevant for normals of edges of polygon faces
- RHCS then polygons whose vertices were ACW will be front facing

Right, up and forward vectors

- RHCS is compatible with Maya
- Usually y-axis is up, z-axis is out and x-axis to the right
- Handedness is relevant to rendering, rotating and cross-product of two vectors
- To switch between the two is tedious : scale all point coordinates and camera-to-world matrix by (1, 1, -1)

The World Coordinate Systems

- Each different coordination system is defined with respect to this
- It defines the origin, x, y, z axis from which the other coordinate systems are defined