Temple Fay’s Transcendental Butterfly Curve in Polar Coordinates

Famous 1989 transcendental polar curve combining exponential, harmonic 4θ, and subharmonic sin⁵((2θ−π)/24) wing modulation.

Expressions & Parameters

Primary Function f₁(x)

Comparison Curve f₂(x)

Plot Tangent Line at x₀ Shade Integral [a, b]

Live Parameter Sliders (a, b, c)

a1.00

b1.00

c0.00

Interactive Coordinate Plane (Drag to Pan · Wheel to Zoom)

(0.000, 0.000)

Pre-Computed Analytical Curve Reference

Polar Lawr(θ) = exp(sin(theta)) - 2*cos(4*theta) + sin((2*theta - pi)/24)^5

Cartesian Mappingx = r(θ)·cos θ , y = r(θ)·sin θ

Sweep Domainθ ∈ [0, 37.699 rad]

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Mathematical Reference & Frequently Asked Questions

How can I modify the parameters of Temple Fay’s Transcendental Butterfly Curve in Polar Coordinates?

Drag sliders a, b, and c in the expression sidebar or edit the formula directly to observe real-time shifts in amplitude, frequency, roots, and extrema.