Coordinate Systems

How do you communicate where you buried the treasure? You can pick a landmark, a direction, and count number of steps. Mathematicians took centuries to realize the potential of this idea!

You can do a lot using just straight edge and a compass… Ancient Greeks were masters at this type of geometry: they were able to prove mathematical results, such as Pythagorean theorem using these simple tools.

But there are situations in which this is not enough: How do we record information about relationships between different shapes?

The Cartesian Plane Choose a point in the plane Draw two perpendicular axes through it, one horizontal and one vertical Any point in the plane can be reached from the origin by travelling a certain distance along horizontal axis and a certain distance along vertical axis

Algebra + Geometry

Function Graphs

Shape Representation 2D polygons are represented by chains of vertices (points) Each pair of neighboring vertices is connected by a straight line Chain may be clockwise or anticlockwise A shape represented numerically can be manipulated mathematically

Area of a Shape

Sum over the results of multiplying an x by the next y, minus the next x by the previous y When the last vertex is selected, it is paired wit the first vertex The result is multiplied by half If the set of vertices is clockwise, the area is negative

Theorem of Pythagoras in 2D

3D Cartesian Coordinates

Polar Coordinates