Four-Petal Rose Tangent-Projection Locus Explorer

Rose r = a cos(2θ)Moving BTangent at BFixed CProjection DLocus of D

Geometry and motion

Rose scale a

Fixed point C: x

Fixed point C: y

Angle θ

Swing speed

Locus resolution

θ is animated by reflected linear motion: −π → +π → −π. A four-petal rose is already fully traced by any interval of length π, but the full swing is retained so the reversing motion is visible.

Current quantities

Moving point B—

Projection D—

Speed |B′(θ)|—

Projection distance CD—

Locus bounding box—

Polar rose: r = a cos(2θ)

Cartesian parametrization:
x(θ) = a cos(2θ)cos θ
y(θ) = a cos(2θ)sin θ

For tangent direction v = (x′,y′), the perpendicular foot of C onto the tangent at B is
D = B + v · ((C−B)·v)/(v·v).

The limiting tangent is used at the origin-crossings, where B′(θ)=0.