A wellmeaning senior citizen once erroneously contended that a "perfect oval" could only be constructed with a straight edge and a pair of compasses.
The theoretical construction that he described so very closely approximates the ellipse given by the equation: (x^2)/16 + (y^2)/9 = 1.
Required:
1. Emulate such a construction
2. Suggest the difference in area of these two entities if this construction and an ellipse representing the above equation are drawn at the same scale (let them share a common major radial length).
To my knowledge, oval and ellipse refer to the same thing, ellipse being the 'technical' term.
Construction of the first quadrant portion
of the "Oval":
Label points O(0,0), A(4,0), and B(0,3).
Construct point M on line segment AB such
that BM = 43 = 1. Construct the
perpendicular bisector of line segment AM
which intersects the xaxis at point C and
the yaxis at point D. Construct circular
arc (starting at point A and ending at
point E on the perpendicular bisector) with
center C and radius CA. Construct
circular arc (starting at point E and
ending at point B) with center D and radius
DB. The rest of the "Oval" is obtained by
reflecting the first quadrant about the
xaxis, the yaxis, and the origin.
Area of "Oval" = 4*[ Area(Sector CAE) +
Area(Sector DEB) 
Area(Triangle OCD)]
= 4*[ (1/2)*(5/2)^2*arcsin(4/5) +
(1/2)*(5)^2*arcsin(3/5) 
(1/2)*(3/2)*(2)]
~= 37.766245665
Area of Ellipse = (PI)*(4)*(3)
~= 37.699111843
(Area of "Oval")(Area of Ellipse)

Area of Ellipse
~= 0.001780780

Posted by Bractals
on 20061118 17:28:31 