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Application of the Pythagorean Theorem to Axis Rotation in
Analytic Geometry
Intended Audience:
Anyone who has taken pre-calculus, having some familiarity with
the distance formula, the equations of conic sections, function
translation and magnification, basic sines and cosines, and
difference identities.
Abstract:
This report displays an old image proof of the Pythagorean
theorem, reviews conic sections, derives an algebraic method for
rotating a graph about the origin, and then gives convincing
examples of the method as well as insights. Rotation axis may
have applications in computer graphics, such as displaying an
elliptical orbit whose axis is not parallel to any screen axis.
Our Goal
Conic Sections
Pythagorean Theorem and Proof
Our Geometric Approach
Derivation
x = sqrt(x 2 + y 2 )cosA
y = sqrt(x 2 + y 2 )sinA
x 0 = sqrt(x 2 + y 2 )cos(A − B)
y 0 = sqrt(x 2 + y 2 )sinA(A − B)
x 0 = sqrt(x 2 + y 2 )[cosAcosB + sinAsinB]
y 0 = sqrt(x 2 + y 2 )[sinAcosB − sinBcosS]
x 0 = xcosB + ysinB
y 0 = ycosB + xsinB
Parabola Example:
y = x2
Rotate: -45 degrees
x 0 = xcosB − ysinB
y 0 = ycosB + xsinB
ycos45 + xsin45 = (xcos45 − ysin45)2
ysqrt(2) − xsqrt(2) = x 2 + 2xy + y 2
Treat x as a constant. Group by powers of y .
Get quadratic equation in y. Use quadratic formula.
Plot the two halves.
Equation of an Axial Hyperbola
Rotating a Familiar Hyperbola
y = x1
xy = 1
Rotate: -45 degrees
x 0 = xcosB − ysinB
y 0 = ycosB + xsinB
(xcos45ysin45)(ycos45 + xsin45) = 1
2x 2 2y 2 = 1
Clearly this is a horizontal hyperbola of half separation sqrt(2).
Conclusions
General equation of a conic section:
Ay 2 + Bxy + Cx 2 + Dy + Ex + F = 0
An xy term indicates the main axis is not parallel to either the x or
y axis.
B 2 − 4AC = 0
means parabola: think only one of the A or C is present in y = x 2
B 2 − 4AC > 0 means hyperbola: think y = x1 : xy = 1 > 0
B 2 − 4AC < 0
means circle or ellipse: think rotation is absent, and x 2 and y 2
have the same sign.
To rotate by B degrees, replace x with xcosB − ysinB
and replace y with xsinB + ycosB.