User:Klunky40

Klagge's Theorem

Although I discovered the formula, A = ½abt, on my own, I have always wondered if others had discovered it before me. I have searched through several texts but only found references to A = ½t for the hyperbolic sector. Recently (the summer of 2008) I did an internet search and found a page that helps you calculate many things about a hyperbola including the area of a sector. This is the first time I have seen “my” formula and I have yet to find it applied to the ellipse as I have done in the following theorem.

The following theorem unifies the area of a sector of a hyperbola, an ellipse and a circle and is stated as follows:

1)   For a hyperbola with a semi-transverse axis of “a”, a semi-conjugate axis “b”, and the parameter t = cosh-1(x/a) = sinh-1(y/b), the area of the hyperbolic sector whose vertices are the point P(x,y) on the hyperbola, and O(0,0) and Q(a,0) is equal to ½abt.

) For an ellipse with a semi-major axis of “a”, a semi-minor axis of “b”, and the parameter t = cos-1(x/a) = sin-1(y/b), the area of the elliptic sector whose vertices are the point P(x,y) on the ellipse, and O(0,0) and Q(a,0) is equal to ½abt.

3) For a circle of radius r, and the parameter t = cos-1(x/r) = sin-1(y/r), the area of the circular sector whose vertices are the point P(x,y) on the circle, and O(0,0) and Q(r,0) is equal to ½abt = ½rrt = ½r2t = ½r2Θ (where a = b = r and Θ = t = the central angle of the sector in circular radians).

When the parameter t = 1 in each of the above cases, “t” can be referred to as a hyperbolic radian, or an elliptic radian or a circular radian, respectively. It must be noted that only the circular radian is proportional to the actual size of the angle when measured with a common protractor.