'============================================================================
'   G.H. George                  EPICYC.BAS                       2001 08 17
'   Traces epicycloids of n cusps from a circle with 120 points.    
'   From SYMmetry plus, #15, Summer 2001 (Mathematical Association), p. 10.
'============================================================================

'   Procedures:
'   Intro()     prints introductory screen
'      x(i)     returns the x coordinate of point i on the circle
'      y(i)     returns the y coordinate of point i on the circle
DECLARE SUB Intro ()
DECLARE FUNCTION x# (i AS INTEGER)
DECLARE FUNCTION y# (i AS INTEGER)

'   Global constants:
'             Pi:   pi
'      numPoints:   number of points on the circle
DEFINT N
CONST Pi = 3.141592653589794#
CONST numPoints = 120

'   Variables:
'   n           number of cusps on the epicycloid
'   i           loop counter = index number of point
'   j           index number of point (n+1) times further along the circle
DIM n AS INTEGER, i AS INTEGER, j AS INTEGER

CALL Intro

SCREEN 8
FOR n = 1 TO 5
    IF n = 1 THEN
        PRINT "Epicycloid of 1 cusp (cardioid)"
    ELSE
        PRINT "Epicycloid of"; n; "cusps"
    END IF

'---------------------------------------------------------------------------
' Set the radius of the guide circle  r = 90 pixels  on the
' 640x200 SCREEN 8  and its centre at (xc, yc) = (320, 100) pixels.
' Each point is at (xi, yi), where
'     xi = xc + r*cos(theta)
'     yi = yc + r*sin(theta)    [evaluated in functions x(i), y(i)]
'     angle theta = (i/numPoints)*360 deg
' The other end of the line is at (xj, yj), (n+1) times further along
' the guide circle, at angle (n+1)*theta, which is equivalent to
'   i -> (n+1)*i MOD numPoints
'---------------------------------------------------------------------------
    FOR i = 1 TO numPoints
        j = ((n + 1) * i) MOD numPoints
        LINE (x(i), y(i))-(x(j), y(j)), 14    ' Yellow lines
    NEXT i

    DO WHILE INKEY$ = "": LOOP      ' Pause until key pressed.
    IF n < 5 THEN CLS
NEXT n

END

SUB Intro
'============================================================================
' Print introductory screen.
'============================================================================

' No procedures or local variables.

SCREEN 0: CLS : COLOR 11, 0             ' Cyan on black
PRINT TAB(25); "Program  Epicyc.bas"
PRINT TAB(25); "         =========="
PRINT
PRINT "From each of"; numPoints; "points on a guide circle, ";
PRINT "a line is drawn"
PRINT "to the point (n+1) times further along the guide circle."
PRINT "The envelope formed by all of these lines is an epicycloid of";
PRINT " n cusps."
PRINT "This is a crude implementation of the algorithm presented by"
PRINT "David Crawford in the article ";
PRINT CHR$(34); "Curves from Straight Lines"; CHR$(34); ", in"
PRINT "SYMmetry plus, #15, Summer 2001 (Mathematical Association), page 10."
PRINT
PRINT "In order to proceed from one value of  n  to the next, ";
PRINT "press any key."
PRINT
COLOR 14, 0                             ' Yellow on black
PRINT "Press any key to continue:  ";

DO WHILE INKEY$ = "": LOOP              ' Pause until any key is pressed.

END SUB

DEFDBL X
FUNCTION x (i AS INTEGER)
'============================================================================
'   Returns the x coordinate of point i on the circle
'============================================================================

'   No procedures or local variables.

' angle theta = (i/numPoints)*360 deg
' xi = xc + r*cos(theta)

x = 320 + 90 * COS((i / numPoints) * 2 * Pi) * 12 / 5
' Aspect ratio correction:  10/5 for screenshot, 12/5 for monitor.

END FUNCTION

DEFDBL Y
FUNCTION y (i AS INTEGER)
'============================================================================
'   Returns the y coordinate of point i on the circle
'============================================================================

'   No procedures or local variables.

' angle theta = (i/numPoints)*360 deg
' yi = yc + r*sin(theta)

y = 100 + 90 * SIN((i / numPoints) * 2 * Pi)

END FUNCTION