Hodograph Turtles

1. Hodograph Turtles Tao Ju, Ron Goldman Rice University

2. Introduction • LOGO • Drawing with FORWARD and TURN • Polygons, stars, … and fractals • Turtle Geometry • Local and coordinate free geometry • Morphing, L-systems, Plant modeling, theory of relativity…

3. Classical Turtle • Turtle state: Position (P), Direction (w) • Turtle commands: • FORWARDd Pnew = P + d w • TURNa w1new= w1 cos(a)- w2 sin(a) w2new= w1 sin(a)+ w2 cos(a) • PEN_UP, PEN_DOWN w P

4. Classical Turtle • Turtle program • Initial state: P = {0,0} and w = {1,0} • Sequence of turtle commands • Plots the trace of positionP Turtle Program Turtle Geometry

5. Hodograph Turtle • Motivation: Plot the trace of directionw • Hodograph: tangential trajectory • Turtle state: Direction (w) • Not affected byFORWARD Command Classical Turtle Hodograph Turtle w w FORWARD 1: Pnew P wnew wnew TURN/6: w w P

6. Classical vs. Hodograph Classical Turtle • Local vs. Global coordinate frame Hodograph Turtle

7. Shapes Inscribed In Circles • Hodograph turtle makes programming easier Rosette Classical Turtle Hodograph Turtle

8. Shapes Inscribed In Circles • Hodograph turtle makes programming easier Circle & Star Classical Turtle Hodograph Turtle

9. Resize • RESIZEs: wnew = s w Program Classical Turtle Hodograph Turtle

10. Fractals – Classical Turtle • Recursive Turtle Program (RTP) • Base case + Recursion body RTP 1 Sierpenski Triangle 0 1 2 3 4 5

11. Fractals – Classical Turtle RTP 2 Sierpenski Triangle

12. Fractals – Hodograph Turtle • Hodograph path helps to • Reveal how the fractal is drawn • Reflect the simple recursive structure Classical Hodograph I Hodograph II

13. Fractals – Hodograph Turtle Classical “Koch Snowflake” Hodograph • New way of generating fractals

14. Fractals – Hodograph Turtle Classical “C-Curve” Hodograph • New way of generating fractals

15. Anchor Commands • Motivation: Free the poor creature (from being tethered to the origin) ! • Augmented hodograph turtle (P’, w) • Draws the trace of ( P’ + w ) • Initial state: P’ = {0,0} • Anchor_Down: P’ stays fixed • Anchor_Up: P’ moves with P

16. Augmented Hodograph Turtle Program Hodograph Aug. Hodograph

17. Anchors and Fractals • The augmented hodograph turtle generates the same fractal in the limit as the classical turtle if : • Both the pen and the anchor are up in the recursion body. • In the base case, the pen is down and either • The anchor is up, or • The anchor is down and the turtle commands introduce no net change in the classical turtle's position vector P.

18. Anchors and Fractals Classical Turtle 1 3 5 Augmented Hodograph Turtle 1 3 5

19. Summary F:FORWARD,T:TURN,P:PEN,A:ANCHOR

20. Summary • Hodograph turtles can • Simplify drawing of shapes inscribed in circles • Reveal how the classical turtle geometry is drawn • Reflect recursive structure of turtle programs • Generate new fractals • As powerful as classical turtles !

21. Open Questions • Extending theories of classical turtle to hodograph turtles • Looping Lemma, Space-time warping, non-conformal mappings, etc. • Easier than classical turtle for teaching? • No FORWARD command • Single transformation: rotation