Deriving The Parameters Of Parabolas Tangent To A Circle
In a previous story, I described how I became fascinated by the properties of a square spiral. In particular, I observed that the average of the position of all points in the spiral converged to 4 partial parabolas arranged around a 1/2 unit square whose bottom left corner was positioned at the origin (0,0).
One property of these partial parabolas is the tangents of adjacent partial parabolas were identical to each other and also to the circle of radius √2/4, centered on (1/4,/1,4) that passes through the corners of the 1/2 unit square.
To further investigate the partial parabolas of this kind, particularly parabolas arrayed around the edges of a circle, I thought it would be useful to generalize the derivation of their characteristic parameters.
To start with, we will derive the equation of a parabola that subtends an arc of ±𝛼 radians of a circle of radius r, centered on the origin. The parabola’s axis of reflection is the x-axis, specifically:
The chief constraint is that at the point of intersection of the circle of radius r, the parabola must be tangent to the circle and thus perpendicular to the circle radius at angle 𝛼 to the x-axis
The coordinates of the point of intersection are:
The slope of the tangent to the circle at angle 𝛼 is:
The slope of the parabola at y (in terms of y) is:
Given the constraint, we have, at y=r.sin(𝛼)
Re-arranging, we get:
Substituting for x, a, y in the equation of the parabola, we get:
Rearranging and simplifying several times:
The working is not shown, but it is possible to express k as a function of the curvature parameter a, as follows:
Conclusion
This article has investigated the parameters of a parabola that intersects tangentially with a circle of radius r, centered on the origin. The point of intersection is the point at which a radius which is elevated at angle 𝛼 from the x-axis intersects with the circumference of the circle
The parameters fit the equation:
The parameters are:
The diagram below shows the parabolas that intersect the unit circle for radii at various 𝛼 = π/n, for n in the range 3 to 12.
Additional Representations
I will save the derivation of this equation for another article, but here is a representation of the partial parabola in terms of polar coordinates:
here r_p and r_c represent the radial coordinates of the parabola and the reference circle respectively, and 𝛼 is the parameter used to derive the equation of the parabola.