Turtle Programming. Ron Goldman Department of Computer Science Rice University

Transcription

1 Turtle Programming Ron Goldman Department of Computer Science Rice University

2 Shapes

3 Turtles What the Turtle Knows CURRENT_POSITION = (x, y) location -- where she is CURRENT_HEADING = (u,v) vector -- where she is going (Slightly more than the average freshman.) Turtle Commands FORWARD(D) -- change position, and draw a straight line of length D ª MOVE(D) -- same as FORWARD without drawing a line TURN(A) -- change heading by A, but do not change position RESIZE(S) -- change step size by a factor of S, but do not change position or direction Usual collection of control commands -- loops, conditionals, K Turtle Programs Finite sequence of FORWARD, TURN, AND RESIZE commands Draws a piecewise linear curve

4 UPDATING POSITION AND HEADING FORWARD(D) -- Translation x new = x old + D u currect y new = y old + D v currect TURN(A) -- Rotation u new = u old cos(a) v old sin(a) v new = u old sin(a) +v old cos(a) RESIZE(S) -- Scaling u new = S u old v new = S v old Conventions D < 0 FORWARD(D) moves the turtle backwards A > 0 TURN(A) rotates the turtle counterclockwise A < 0 TURN(A) rotates the turtle clockwise S < 0 RESIZE(S) rotates the turtle by 180 o

5 UPDATING POSITION (x new, y new ) (u,v) D (x old, y old ) (u,v) x new = x old + D u currect y new = y old + D v currect

6 UPDATING DIRECTION (u new,v new ) = ( r cos(θ +α), rsin(θ +α) ) r (u old,v old ) = ( r cos(θ), rsin(θ) ) α θ r r cos(θ) rsin(θ)

7 UPDATING DIRECTION Polar Coordinates u new = r cos(θ +α) v new = r sin(θ +α) u old = r cos(θ) v old = r sin(θ) Trigonometric Identities cos(θ +α) = cos(θ)cos(α) sin(θ)sin(α) sin(θ +α) = sin(θ)cos(α)+ cos(θ)sin(α) Substitution u new = r cos(θ)cos(α) rsin(θ)sin(α) = u old cos(α) v old sin(α) v new = r sin(θ)cos(α)+r cos(θ)sin(α) = v old cos(α)+ u old sin(α) Matrix Formulation ( u new v new ) = u old v old ( ) cos(α) sin(α) sin(α) cos(α)

8 UPDATING DIRECTION (Revisited) w w new P α wcos(α) w sin(α) w w new = w cos(α)+ w sin(α) w = (u,v) w = ( v,u) u new = ucos(α) vsin(α) v new = vcos(α)+ usin(α)

9 Scaling (u new,v new ) S (u old,v old ) u new = S u old ( u new v new ) = ( u old v old ) S 0 v new = S v old 0 S

10 UPDATING POSITION AND HEADING FORWARD(D) -- Translation ( x new y new ) = x old y old ( ) + D ( u currect v currect ) TURN(A) -- Rotation ( u new v new ) = u old v old ( ) cos( A) sin(a) sin( A) cos( A) RESIZE(S) -- Scaling ( u new v new ) = u old v old 0 S ( ) S 0 TURNandRESIZE(A,S) -- Rotation and Scaling S cos(a) Ssin(A) ( u new v new ) = ( u old v old ) S sin(a) S cos(a)

11 Communication Human to Turtle FORWARD, MOVE, TURN, and RESIZE commands only Illegal to use coordinates High Level Programming Language LOGO = Programming Language for Geometry Turtle to Computer Coordinate Computations Low Level Programming Language Coordinates = Assembly Language for Geometry

12 Computational Geometry Themes Interaction between geometry and programming Can investigate geometry by studying turtle programs Can understand turtle programs by investigating geometry Motivation To free ourselves from coordinates To think and to program at a higher level Goals Fractals Transformations

13 Comparison Turtle Geometry Procedures Local Intrinsic Coordinate Geometry Equations Global Extrinsic

14 Turtle Procedures Simple Shapes Polygons Stars Circles Spirals Wheels Rosettes M Higher Level Procedures Shift Spin Scale

15 Observations FORWARD, SHIFT, and SCALE are 1-Dimensional boring shapes straight line motion TURN and SPIN are 2-Dimensional interesting patterns circular motion

16 Regular Polygons Program Polygon(L=Length of Side, N=Number of Sides) Repeat N [FORWARD(L), TURN(360/N)] Observation Angle = Exterior Angle

17 Stars Program Polygon(L=Length of Side, A=Angle) Repeat N [FORWARD(L), TURN(A)] Observation Angle = 72 Pentagon Angle = 144 Star

18 Circles Program Circle Repeat [FORWARD(1), TURN(1)] Questions What happens if we change the FORWARD parameter? What happens if we change the TURN parameter?

19 Spirals Iterative Program Spiral(Length, Angle, ScaleFactor) REPEAT N TIMES FORWARD(Length) TURN(Angle) RESIZE(ScaleFactor) Recursive Program Spiral(Length, Angle, ScaleFactor) If (Length > Max_Length) STOP FORWARD(Length) TURN(Angle) Spiral(Length ScaleFactor, Angle, ScaleFactor)

20 Shift, Spin, and Scale SHIFT Turtle Program REPEAT N TIMES Turtle Program MOVE D {Note the use of MOVE instead of FORWARD} SPIN Turtle Program REPEAT N TIMES Turtle Program TURN A SCALE Turtle Program REPEAT N TIMES Turtle Program RESIZE S