Elllipses in BBC Basic

Ellipses in BBC Basic

We start with the circle program previously discussed:

MODE8:OFF
      ORIGIN 640,512
      R=300
      PLOT69,0,0 : REM mark the centre
      MOVE R,0
      FOR t=0 TO 360
x=R*COSRADt
y=R*SINRADt
        DRAW x,y
        delay=INKEY(1)
      NEXT

We can modify this quite easily to draw an ellipse:

      MODE8:OFF
      ORIGIN 640,512
      a=500 : b= 300
      PLOT69,0,0 : REM mark the centre
      MOVE a,0
      FOR t=0 TO 360
        x=a*COSRADt
        y=b*SINRADt
        DRAW x,y
        delay=INKEY(1)
      NEXT

Here a represents the semi-major axis (half the ellipse's longer axis) lying here along the x-axis
and b represents the semi-minor axis (half the ellipse's shorter axis) lying here along the y-axis

In maths books, the representation of an ellipse is often given in terms of the ellipticity of the ellipse, e.where

            b = a * SQR(1 - e^2), where * is a multiplication sign, SQR specifies a square root and e^2 specificies e to the power two or "e squared".

(e is zero for a circle, unity for a parabola and between these two value for an ellipse).

The two foci of the ellipse lie at a distance of a*e either side of the the centre point on the major axis. In order to represent a planetary orbit we need to move the ellipse along the x-axis by a*e in order to bring a focus to the origin which now signifies the position of the sun:

      MODE8:OFF
      ORIGIN 140,512
      a=500 : b= 300 : e = SQR(1-(b/a)^2)
      PLOT69,0,0
REM mark the origin, now also one focus of ellipse
      MOVE a+(a*e),0
      FOR t=0 TO 360
        x=(a*COSRADt) + (a*e)
        y=b*SINRADt
        DRAW x,y
        delay=INKEY(1)
      NEXT

If we now wish to show the ellipse at any angle,T (measured clockwise) from the horizontal (x-axis) we need to rotate each point of the ellipse about the origin:

      MODE8:OFF
      ORIGIN 140,700
      a=500 : b= 300 : e = SQR(1-(b/a)^2) : T=20
      PLOT69,0,0
      REM mark the origin, now also one focus of ellipse
      FOR t=0 TO 360
        x=a*COSRADt
        y=b*SINRADt
        REM shift focus to origin
        xx=a*COSRADt + (a*e)
        yy=y
        REM rotate all points by T degrees (clockwise)
        xxx=(xx*COSRADT) + (yy*SINRADT)
        yyy=-(xx*SINRADT) + (yy*COSRADT)
        IF t=0 MOVE xxx,yyy
        DRAW xxx,yyy
        delay=INKEY(1)
      NEXT

Having got to this point we are now well equipped to represent planet and satellite orbits about their central bodies (sun, earth etc).

For more ellipse demos click here

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