Rolling Circles along curves

The base curve is simple in our case---it is just a horizontal line:

1. The center of the circle C moves horizontally. Because it rolls without slipping, the horizontal distance the center travels is exactly equal to the arc length rolled, which is rθ.
2. The coordinates of the center C at any angle θ are (rθ, r).

• The point P rotates backward relative to the center.
• Measuring from the lowest point of the circle (the contact point), the vector from the center C to the point P points downward and backward.

• The relative horizontal displacement is −r sinθ (see right triangle ▲CQP with angle ∡C = θ in Figure 1.2).
• The relative vertical displacement CQ is −r cosθ.

• Cusps: The curve touches the flat baseline (y = 0) at periodic intervals where θ = 2πn, independently of the radius r.
• Peaks: The maximum height of the curve is 2r, occurring at θ = π, 3π, ….
• Applications: The cycloid solves both the brachistochrone problem (the path of fastest descent under gravity) and the tautochrone problem (where the time taken by an object sliding down the curve is independent of its starting point).

To find the distance the tracing point actually travels during one full rotation (0 to 2π), we use the parametric arc length formula:

1. Copy and paste the script directly into a blank Mathematica notebook.
2. Select the cell and press Shift + Enter to run it.
3. Mathematica will generate the frames locally and save a high-quality file named cycloid_animation.gif directly inside your current working directory (you can find or change this path using the command Directory

Let a stationary base circle have a radius of R centered at the origin (0,0). A smaller rolling circle with a radius of r rolls along its outer circumference without slipping. We trace a point P starting on the outer rim where the two circles touch on the positive x-axis. As the rolling circle moves through an angle θ around the center of the base circle:

1. The center C(x_c, y_c) of the moving circle is located at a distance of R + r from the origin. Its coordinates are:
   \[ x_c = \left( R + r \right) \cos\theta , \qquad y_c = \left( R + r \right) \sin\theta . \]
2. Because it rolls without slipping, the arc length traveled on both circles must be equal (Rθ = rα, where α is the internal rotation angle of the rolling circle). The position of P(x, y) relative to the center C(x_c, y_c) depends on the total accumulated angle of rotation of the rolling radius, which is
   \[ \phi = \theta + \alpha = \theta + \frac{R}{r}\,\theta = \left( \frac{R + r}{r} \right) \theta . \]
To understand how the latter formula connecting ϕ and θ is derived, we need to look closely at the condition for rolling without slipping and track the angles from two different points of view: an observer standing outside the system (the fixed base frame) and the moving center C of the rolling circle.

Let’s set up the system with a large stationary base circle of radius R centered at the origin, and a smaller circle of radius r rolling along its outside edge.

• θ: The angle of the center of the rolling circle relative to the origin. This is how far the small circle has traveled around the large circle.
• The point of contact on the large circle has traveled an arc length of:
  \[ s_{\mbox{base}} = R\,\theta . \]
Because the small circle rolls without slipping, the length of the arc it has rolled out along its own circumference must be exactly equal to the distance it covered on the large base circle. If we let α be the angle that the small circle has rotated relative to the moving line connecting the two centers, its internal arc length is:    s_rolling = rα.

Equating the two arc lengths, we get

\[ R\theta = r\alpha \qquad \Longrightarrow \qquad \alpha = \left( \frac{R}{r} \right) \theta . \]
The angle α only measures how much the circle has spun relative to the line connecting the centers. However, that center line itself is also rotating around the origin at an angle of θ.

To find the total, absolute rotation angle ϕ of the tracking point relative to the fixed horizontal x-axis (the base frame), we have to add these two rotations together:

\[ \phi = \theta + \alpha = \theta + \left( \frac{R}{r} \right) \theta = \left( \frac{R+r}{r} \right) \theta . \]

Combining the moving center coordinates with the trigonometric rotation adjustments gives the parametric equations for an epicycloid:

\[ \begin{split} x (\theta ) &= \left( R + r \right) \cos\theta - r\,\cos \left( \frac{R+r}{r}\,\theta \right) , \\ y (\theta ) &= \left( R + r \right) \sin\theta - r\,\sin \left( \frac{R+r}{r}\,\theta \right) . \end{split} \]
The following Mathematica code creates a cardioid (a heart-shaped epicycloid) by setting R = 2 and r = 1 (a 2:1 ratio). You can easily change these parameters at the top of the script to create different shapes (e.g., setting R = 2, r = 0.5 creates a nephroid).
(* 1. Define Radii Parameters (R = base circle, r = rolling circle) *) R = 2; r = 1; (* Custom Epicycloid Equations *) epicycloidX[t_] := (R + r)*Cos[t] - r*Cos[((R + r)/r)*t]; epicycloidY[t_] := (R + r)*Sin[t] - r*Sin[((R + r)/r)*t];k (* 2. Build the Frame List *) frames = Table[ Show[ (* Static Background Base Circle & Full Path Outline *) ParametricPlot[{R*Cos[t], R*Sin[t]}, {t, 0, 2 Pi}, PlotStyle -> Directive[baseCircleColor, Thickness[0.004]], PlotRange -> {-(R + 2 r) - 0.5, (R + 2 r) + 0.5}, Axes -> False, AspectRatio -> Automatic, Background -> GrayLevel[0.98], ImageSize -> Large ], (* Overlay a faint ghost line of the total expected epicycloid *) ParametricPlot[{epicycloidX[t], epicycloidY[t]}, {t, 0, 2 Pi}, PlotStyle -> Directive[GrayLevel[0.85], Dashed] ], (* Active Growing Epicycloid Curve *) ParametricPlot[{epicycloidX[t], epicycloidY[t]}, {t, 0, Max[0.001, maxT]}, PlotStyle -> Directive[traceColor, Thickness[0.007], CapForm["Round"]] ], (* Dynamic Visual Components *) Graphics[{ (* Moving Center of Rolling Circle *) With[{cX = (R + r)*Cos[maxT], cY = (R + r)*Sin[maxT]}, { (* The rolling circle body *) Directive[rollingColor, Thickness[0.004]], Circle[{cX, cY}, r], Directive[Opacity[0.08], rollingColor], Disk[{cX, cY}, r], (* Radial spoke line connecting roller center to trace point *) Directive[rollingColor, Thickness[0.003]], Line[{{cX, cY}, {epicycloidX[maxT], epicycloidY[maxT]}}], (* Roller center coordinate point *) rollingColor, PointSize[0.015], Point[{cX, cY}] }], (* The main active tracing point on the rim *) traceColor, PointSize[0.022], Point[{epicycloidX[maxT], epicycloidY[maxT]}] }] ], {maxT, 0, 2 Pi, 2 Pi/100} ]; (* 3. Export to Seamless Loop GIF *) Export["epicycloid_animation.gif", frames, "AnimationRepetitions" -> Infinity, "DisplayDurations" -> 0.04 ]

To clarify the geometry first: an epicycloid with R = r creates a cardioid (1 cusp), while an epicycloid with R = 2r creates a nephroid (2 cusps). Below are two options for Wolfram Mathematica code. The first plots both curves statically side-by-side, and the second creates an interactive animation showing the transition from the cardioid to the R = 2r curve.

(* Define the general epicycloid equations *) x[R_, r_, t_] := (R + r) Cos[t] - r Cos[((R + r)/r) t]; y[R_, r_, t_] := (R + r) Sin[t] - r Sin[((R + r)/r) t]; rVal = 1; (* Plot Cardioid (R = r) *) plotCardioid = ParametricPlot[ {x[rVal, rVal, t], y[rVal, rVal, t]}, {t, 0, 2 Pi}, PlotStyle -> Directive[Thick, Blue], PlotLabel -> "Cardioid (R = r)", Axes -> True, GridLines -> Automatic ]; (* Plot Epicycloid (R = 2r) *) plotNephroid = ParametricPlot[ {x[2 rVal, rVal, t], y[2 rVal, rVal, t]}, {t, 0, 2 Pi}, PlotStyle -> Directive[Thick, Red], PlotLabel -> "Epicycloid (R = 2r)", Axes -> True, GridLines -> Automatic ]; (* Display together *) GraphicsRow[{plotCardioid, plotNephroid}, ImageSize -> Large]
This code uses Manipulate to let you smoothly change the value of R from r (cardioid) to 2r, showing exactly how the shape transforms.
(* Interactive slider morphing from Cardioid to R = 2r *) Manipulate[ Module[{r = 1}, ParametricPlot[ {(R + r) Cos[t] - r Cos[((R + r)/r) t], (R + r) Sin[t] - r Sin[((R + r)/r) t]}, {t, 0, 2 Pi}, PlotStyle -> Directive[Thick, Purple], PlotRange -> {{-4, 4}, {-4, 4}}, Axes -> True, GridLines -> Automatic, PerformanceGoal -> "Quality" ] ], {{R, 1, "Fixed Radius (R)"}, 1, 2, Appearance -> "Labeled"} ]
How to use this code in Mathematica:
• Copy either block of code.
• Paste it directly into a clean cell in a Mathematica Notebook.
• Press Shift + Enter to evaluate and generate the graphics.



   ■
End of Example 2

Mathematically, the center of the circle C that touches the ellipse externally is calculated as

Now we plot the same ellipse together with some pieces of circles that touches the ellipse outside.

Now we plot circles of fixed radius touching the ellipse outside:

For an ellipse defined parametrically by:

We want to add a trace line to the plot that shows the continuous path the red dot/point carves out as it rolls all the way around. While a unit circle rolls around the ellipse, the red dots will trace a physically accurate cycloid-like progression that reflects the true geometry of the ellipse.

For a regular planar curve given by y = f(x), the curvature κ at any point x is calculated using its first and second derivatives:

For 2D curve r(t) = (x(t), y(t), we compute the unit tangent vector defined as

Archimedes (c. 287 – c. 212 BC) is celebrated as the supreme mathematical and physical genius of antiquity. Operating with cumbersome Greek numerals and limited prior tools, he single-handedly derived foundational concepts for calculus, mechanics, and geometry.

The Archimedean spiral (also known as Archimedes' spiral, the arithmetic spiral) is a spiral curve described in polar coordinates by the equation r = 𝑎 + bθ. In Cartesian coordinates (x, y), the parametric equations with respect to the parameter θ (the angle of rotation in radians) are:

• Logarithmic Spiral (Equiangular Spiral) expands exponentially, meaning its shape stays identical as it grows (self-similarity). The logarithmic spiral was first described by René Descartes in 1638. Its equation in polar is
  \[ r = a\, e^{b\,\theta} . \]
  (* 1. Logarithmic Spiral *) p1 = ParametricPlot[{Exp[0.15 t] Cos[t], Exp[0.15 t] Sin[t]}, {t, 0, 4 Pi}, PlotStyle -> Directive[Thickness[0.01], Blue], PlotLabel -> "Logarithmic Spiral\n(r = \!\(\*SuperscriptBox[\(e\), \(0.15\ \(\t\ heta\)\)]\))", Axes -> True, Ticks -> None]

  

• Fermat's Spiral (Parabolic Spiral) shares a close relationship with the Archimedean spiral but expands much more slowly because the radius depends on the square root of the angle.
  \[ r = \pm a\,\sqrt{\theta} . \]
  (* 2. Fermat's Spiral (Both branches: positive and negative r) *) p2 = ParametricPlot[ {{Sqrt[t] Cos[t], Sqrt[t] Sin[t]}, {-Sqrt[t] Cos[t], -Sqrt[t] Sin[t]}}, {t, 0, 6 Pi}, PlotStyle -> Directive[Thick, RGBColor[0, 0.6, 0]], PlotLabel -> "Fermat's Spiral\n(\!\(\*SuperscriptBox[\(r\), \(2\)]\) = \(\theta\))", Axes -> True, Ticks -> None ];

  

• Hyperbolic Spiral (Reciprocal Spiral) acts as the inverse of an Archimedes' spiral. Instead of moving outward as the angle increases, it starts infinitely far away and winds inward toward the origin.
  \[ r = \frac{a}{\theta} . \]
  p3 = ParametricPlot[{(2/t) Cos[t], (2/t) Sin[t]}, {t, 0.1, 4 Pi}, PlotStyle -> Directive[Thickness[0.01], Orange], PlotLabel -> "Hyperbolic Spiral\n(r = 2/\(\theta\))", Axes -> True, Ticks -> None, PlotRange -> {{-3, 3}, {-3, 3}}]

  

• Lituus Spiral is similar to the hyperbolic spiral, it originates at infinity and spirals inward, but its radius shrinks even faster.
  \[ r = \frac{a}{\sqrt{\theta}} . \]
  p4 = ParametricPlot[{(2/Sqrt[t]) Cos[t], (2/Sqrt[t]) Sin[t]}, {t, 0.1, 6 Pi}, PlotStyle -> Directive[Thickness[0.01], Red], PlotLabel -> "Lituus Spiral\n(\!\(\*SuperscriptBox[\(r\), \(2\)]\) = 4/\(\t\ heta\))", Axes -> True, Ticks -> None, PlotRange -> {{-4, 4}, {-4, 4}}]

  

• Clothoid (Euler Spiral / Cornu Spiral) is defined by a fundamental geometric property: its curvature increases linearly with its arc length. This means as you travel along the curve, it turns tighter and tighter at a perfectly constant rate. The cleanest definition of the Euler spiral is given by its intrinsic equation, which relates its curvature (κ) directly to its arc length (s):
  \[ \kappa = \frac{s}{a^2} . \]
  In Cartesian coordinates (x, y), we integrate the intrinsic property. This yields equations defined by the Fresnel Integrals:
  \[ s (t) = a \int_0^t \,\cos \left( \frac{u^2}{2} \right) {\text d}u , \qquad y(t) = a \int_0^t \,\sin \left( \frac{u^2}{2} \right) {\text d}u . \]
  (*5. Euler Spiral (Clothoid)*)p5 = ParametricPlot[{FresnelC[t], FresnelS[t]}, {t, -5, 5}, PlotStyle -> Directive[Thickness[0.01], Purple], PlotLabel -> "Euler Spiral (Clothoid)\n(Fresnel Integrals)", Axes -> True, Ticks -> None]

  

1. Graph the curvature of an ellipse given by the position vector field r(t) = (4 cost, 7 sint) for t ∈ [0, 2π].
2. Graph the curvature of the logarithmic spiral given by the position vector field r(t) = (e^−0.2tcost, e^−0.2tsint) for t ∈ [0, 2π].
3. Verify that a circle of radius r, defined by the position vector field x(t) = (rcost, rsint) has curvature κ = 1/r.
4. Compute the curvature of the helix defined by x(t) = (cost, sint, t) in 3D.
5. Roll a circle of radius ⅕ in a counterclockwise fashion around the interior of the ellipse r(t) = (3 cost, 4 sint) for t ∈ [0, 2π].
6. Roll a circle of radius ⅙ in a counterclockwise fashion around the interior of the ellipse r(t) = (3 cost, 4 sint) for t ∈ [0, 2π].
7. Plot the original function y = sin(x) together with its curvature function.
Roll a circle of radius ⅙ in a clockwise fashion around the interior of the ellipse r(t) = (3 cost, 4 sint) for t ∈ [0, 2π].