Maths - 2D Geometry

Abstract

This page is about how geometric algebra can represent objects in two dimensions and some useful concepts for analysing 2D geometry.

To see how the same concepts can be described using more conventional vector algebra see this page.

This page assumes an understanding of coordinate systems described here, you may also want to review angles.

Points and Lines from the Origin

We can use a 2 dimensional vector to represent a displacement from the origin and therefore specify a given point by showing its displacement from the origin. For instance, the point shown below is 5 up and 9 right, which can be represented by the vector (9,5). By convention the x coordinate is shown first, then the y coordinate.

vector in 2 dimentions

Angles

In two dimensions angles are scalar quantities.

angle between 2D vectors

If we have two 2D vectors (V1 and V2) then we could work out the angle between them, at the origin, by using:

angle = acos(real(v1*v2))

where:

or

angle = atan2(v2.y,v2.x) - atan2(v1.y,v1.x)

depending on whether we want to make the angle negative if V1 and V2 are swapped over. This is more fully explained on this page.

The first version uses the vector dot product '•' which generates a scalar from two vectors as explained here. The second version uses the x and y components of the vectors.

If we want to work out an angle at a point other than the origin, then we can translate the point to the origin by subtracting the displacement from all the vectors.

Areas

In two dimensions areas are scalar quantities. An area can be represented by one number, this does not say anything about the shape of the two dimensional object just the amount of 'surface' it contains.

angle between vectors in 2 dimentions

Any two dimensional closed shape encloses an area, this might be a simple shape like a rectangle or a circle or it could be a very complex shape. In any case its area is a single scalar number.

As an example let us take the area enclosed by two vectors as shown in yellow in the diagram above, this area is given by:

area = psudoScalar(v1 * v2)

where:

This is the green area in the diagram below:

angle between vectors in 2 dimentions

Any idea how I can show, geometrically using these diagrams, green area is the same as the yellow area?

Where next

To see how these objects can be projected onto other objects read this page.

To extend these ideas to three dimensions see this page.


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see also:

 

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.

 

Terminology and Notation

Specific to this page here:

 

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