Earlier this week I was doing some work on a tool I'm currently building in stealth (no worries, checkout Tinker Quest).

I thought of implementing a mini drawing tool that works programmatically. The idea was to have something like Python's Turtle module but without much functionality for a start. Honestly I struggled a little, but I eventually figured the math in order to do this sort of thing.

The aim of this post is to share with you how it works and how I achieved this.

A mini command language

The application has a mini command language that directs the tool where to plot the points on a virtual coordinate system and then link them.

Here is an example

M 200 200|R 90|F 100|R 90|F 100|R 90|F 100|R 90|F 100|

As you can see it is pretty simple. Briefly:

• M: Positions the cursor at the coordinates (200, 200) relative to the origin of your screen.
• R: Rotates the point's vector anti-clockwise by units.
• F: Moves the point by units in the direction given by the previous command, R.

The maths

It stars off with a point vector, , where has the value (or whatever coordinate you choose to place the point originally). Moreover, the point vector is thought to point toward the right i.e ->.

That basically means that if we run the above command, these will happen:

• Place the cursor at
• Rotate the point by (the vector is now pointing upwards)
• Move the point by 100 units upward
• Rotate the point by (we are now pointing right)
• Forward by 100 units
• ... and so on.

We end up with a square.

The accumulated rotations are kept in memory though. So, every rotation is added to a variable accAngle and when the above command has been completely executed, the value of accAngle will be 360.

But how is the new point calculated on every iteration? That's where trigonometry comes in.

Say we have a triangle like the one below. We want to move a certain distance forward, whatever the direction of our vector is, the movement corresponds to the displacement in the direction of our point vector, .

Considering our current point, , as the origin, if we compute the values of and , we will get the , displacement we can add to our original vector.

That means to get the new point, we will have where is the distance on x,y from the triangle.