Parametric Speed & Arc Length
Interactive parametric curve visualizer showing position, velocity vector, instantaneous speed |v(t)|, and arc length via Simpson integration. Animate particles along Cycloid, Ellipse, Lissajous, and Butterfly curves with real-time distance accumulation.
Polar Tangent Lines & Slope
Visualize dy/dx for polar curves r=f(θ) using the chain rule conversion to Cartesian. Shows tangent lines, radial vectors, and identifies horizontal/vertical tangents on Cardioid, Rose, Limaçon, and Lemniscate curves.
Vector Curvature & Acceleration
Vector-valued function analyzer showing velocity (tangent), acceleration vectors, curvature κ, and osculating circle with radius R=1/κ. Explore Parabola, Sine Wave, Spiral, and Astroid trajectories with animated particle motion.
Polar Area Between Curves
Polar coordinate area calculator for enclosed regions between two curves. Computes ½∫(R²−r²)dθ via Simpson's rule with adjustable α and β bounds. Visualizes Circle vs Cardioid, Rose vs Circle, Limaçon vs Circle, and dual Rose comparisons.