Hodograph Turtles

Tao Ju, Ron GoldmanRice University

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…

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

Classical Turtle

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

wP PnewFORWARD

1:

TURN /6:P w

wnew

wwnew

w

Command Classical Turtle Hodograph Turtle

Classical vs. Hodograph

Local vs. Global coordinate frame

Hodograph Turtle

Classical Turtle

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

Sierpenski Triangle0

3 4 5

21

RTP 1

Fractals – Classical Turtle

Sierpenski TriangleRTP 2

Fractals – Hodograph Turtle

Hodograph path helps to Reveal how the fractal is drawn Reflect the simple recursive structure

Hodograph IClassical 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 FractalsClassical Turtle

Augmented Hodograph Turtle

1 3 5

1 3 5

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 Arbitrary

Coordinate Frame

Local Global HybridF: 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