Looking at things from a fresh Angle.

• Radians vs Degrees
  • We are all comfortable that a circle can be divided into $360^\circ$
  • But we also know that the c++ trig functions want radians.
  • What is a radian?
    • wikipedia reference
    • This is the SI unit for measuring an angle.
    • It is a dimensionless unit,
    • An angle is measured between 0 and $2\pi$ radians.
  • Converting:
    • $2\pi $ radians = $360^\circ$
    • or $\frac{2\pi}{360^\circ}$
    • which simplifies to $\frac{\pi}{180^\circ}$
    • So
      • $45^\circ = 45^\circ \times \frac{\pi}{180^\circ} $
      • $ = \frac{45\pi}{180}radians $
      • $ = \frac{\pi}{4}radians $
    • Remember Math.PI is a constant in javascript.
  • We need this because all trig functions take radians.
• But I don't know trig functions.
  • Ok, but well....
  • The functions cos and sin are used to compute the relationship between various angels of a right triangle.
    • Given a right triangle (or a triangle with a $90^\circ$ angle.
    • (Wikipedia)
  • The nice part is that this gives us an easy way to go around a circle.
  • (Wikipedia)
    • $\cos(\theta) = \frac{adjacent}{hypotenuse}$
    • $\sin(\theta) = \frac{opposite}{hypotenuse}$
    • $\tan(\theta) = \frac{adjacent}{opposite}$
    • sin and cos are great functions
  • (Wikipedia again)
  • Math.cos(), Math.sin(), ... all exist in javascript.
• These are the basis for Polar Coordinates
  • The is an alternative way to describe a plane.
  • Alternative to Cartesian.
  • r is a distance from the origin
  • $\Theta$ is an angle from the "x" axis.
  • $x = r\cos(\Theta))$
  • $y = r\sin(\Theta))$
  • A demo
• Problem: Draw a 5 sided star centered on the canvas.
  • Note that all of the points on a star are on a circle.
  • Draw a star on the board.
  • And that they are evenly spaced out.
  • Can we find these points?
    • Well the first is at 0 radians
    • The others should be spaced out at $\frac{\pi}{5}$ radians.
  • This is a take on a larger problem, A n/m star polygon
    • n is the number of points (5)
    • m is the skip factor between points (2)
    • A 5/1 star polygon is the pentagon
    • A 5/2 star polygon is the star.
    • A 6/2 or 6/3 is problematic
  • Look here