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

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…

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

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

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

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

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

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

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

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

Fractals – Classical Turtle RTP 2 Sierpenski Triangle

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

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

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

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

Augmented Hodograph Turtle Program Hodograph Aug. Hodograph

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.

Anchors and Fractals Classical Turtle Augmented Hodograph Turtle 1 3 5

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

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 !

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