1. Tao Ju, Ron Goldman Rice University
Hodograph Turtles
Tao Ju, Ron Goldman
Rice University

2. Introduction LOGO Turtle Geometry 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:
FORWARD d
Pnew = P + d w
TURN a
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 position P
Turtle Program
Turtle Geometry

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

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

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
RESIZE s: wnew = s w
Program
Classical Turtle
Hodograph Turtle

10. Fractals – Classical Turtle
Recursive Turtle Program (RTP)
Base case + Recursion body
RTP 1
Sierpenski Triangle 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 Augmented Hodograph Turtle 1 3 5

19. Augmented Hodograph turtle
Summary
Classical Turtle
Hodograph Turtle
Augmented Hodograph turtle
State
P, w w
P’,w
Commands
F,T,P
T,P
F,T,P,A
Geometry
Arbitrary
Concentric
Coordinate Frame
Local
Global
Hybrid
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