Threaded Mode | Linear Mode

Martin Hepperle · (This post was last modified: 04-18-2016 08:07 AM by Martin Hepperle.)

... and for illustration ... a Series-80 BASIC implementation for drawing a curve at "constant speed". You supply the two functions for x(p) and y(p) at the end of the code.

Besides the pure plotting data some additional values are calculated (e.g. the arc length of the curve). For speed you could strip some code and I also have left in some duplicated subexpressions for clarity.

The ticks option can be used to draw monorail tracks if you want to describe the track layout mathematically...

Code:

10 ! HP-85

20 ! Constant speed plotter

30 ! and path length integrator

40 ! for parametric function

50 ! Martin Hepperle, 2016 

60 !

70 ! initial values  

80 T=0 ! time

90 S=0 ! distance

100 P=0 ! parameter at time T

110 V=.5 ! velocity (const.)

120 ! integration parameters

130 T0=.1 ! time step

140 C0=.01 ! fin. diff. step

150 P9=4*PI ! end of parameter

160 ! distance for dt (const.)

170 S0=V*T0

180 C2=2*C0 ! speed up calc.

190 ! -- isotropic plotting --

200 GCLEAR

210 SCALE -(1.1*RATIO),1.1*RATIO,-1.1,1.1

220 MOVE FNX(P),FNY(P)

230 ! integration loop       

240 ! derivative dx/dp

250 D0=(FNX(P+C0)-FNX(P-C0))/C2

260 ! derivative dy/dp

270 D1=(FNY(P+C0)-FNY(P-C0))/C2

280 ! calculate dp

290 ! required for given dt

300 P0=S0/SQR(D0^2+D1^2)

310 ! -- plotting --

320 DRAW FNX(P),FNY(P)

330 ! --- ticks (optional)

340 ! derivative dy/dx

350 D2=D1/D0

360 F=ATN(D2)

370 F0=SIN(F)*.03

380 F1=COS(F)*.03

390 MOVE FNX(P)-F0,FNY(P)+F1

400 DRAW FNX(P)+F0,FNY(P)-F1

410 MOVE FNX(P),FNY(P)

420 ! --- end of ticks

430 ! advance to next

440 P=P+P0 ! parameter

450 T=T+T0 ! time

460 S=S+S0 ! distance so far

470 ! check for termination

480 IF P<=P9 THEN 250

490 PRINT "End Time  t=";T

500 PRINT "Distance  s=";S

510 PRINT "Parameter p=";P

520 ! accuracy check:

530 ! circle: sin(p),cos(p)

540 ! S/PI should be 2.0

550 PRINT "s/PI=";S/PI

560 END

570 ! ----------------

580 ! function x(p)

590 DEF FNX(P)

600 FNX=SIN(P)

610 FN END

620 ! ----------------

630 ! function y(p)

640 DEF FNY(P)

650 FNY=COS(P*.5)

660 FN END

670 ! ----------------

680 END

Of course, one could replace \(arctan()\), \(sin()\) and \(cos()\) by \(sqrt()\) expressions and simplify further.
Then you could replace

Code:

350 ! deleted

360 ! deleted

370 F0=D1*P0/S0*.03

380 F1=D0*P0/S0*.03

which leaves further room for simplification (common subexpressions ) but you must be aware of cases where \(dx/dp=0\) or \(dy/dp=0\).
The implementation will also hang if both derivatives are zero at the same time (singular point).

Martin