A point moving through space carries a frame with it. Tangent, normal and binormal, the circle that best fits a curve, and Kepler’s planets
A moving point traces a curve r(t), and its derivatives are its velocity and acceleration. From them come two numbers that describe the shape of any curve in space, whatever the parametrisation: how sharply it bends, and how fast it twists out of its own plane.
Ride along a space curve with three arrows: tangent, normal and binormal. Switch curves, scrub along them, and turn on the circle that hugs the curve best.
r(t) = (1.5 cos t, 1.5 sin t, 0.3t)
T, N and B are unit vectors; the curve is played at a constant rate in t, so the speed readout changes where the parametrisation hurries or dawdles. Derivatives are computed by five-point finite differences. For this helix the exact values are κ = a/(a² + b²) = 0.641 and τ = b/(a² + b²) = 0.128.
For a curve r(t) the unit tangent is T = r′/|r′|. Measured against arc length s, the rate at which T turns is the curvature: κ = |dT/ds|. Since T has constant length, dT/ds is perpendicular to T, and its direction is the principal normal N. The binormal B = T × N completes a right-handed frame. In terms of any parameter, κ = |r′ × r″|/|r′|³ and the torsion is τ = (r′ × r″)·r‴/|r′ × r″|².
The frame moves according to the Frenet–Serret formulas, found independently by Jean Frenet (1847) and Joseph Serret (1851): dT/ds = κN, dN/ds = −κT + τB, dB/ds = −τN. The fundamental theorem of space curves says κ(s) and τ(s), with κ > 0, determine the curve up to a rigid motion. A curve with τ = 0 everywhere lies in a plane; one with constant κ and τ both non-zero is a helix.
The osculating circle (from the Latin osculari, to kiss) is the circle in the plane of T and N with radius 1/κ, centred at r + N/κ. It matches the curve’s position, direction and bending at the point: of all circles, it agrees with the curve to second order. For the helix (a cos t, a sin t, b t) the formulas give κ = a/(a² + b²) and τ = b/(a² + b²).
Drive a car along a road. The circle at the car is the one that fits the road best right there, and its centre traces a second curve, the evolute.
The car moves at constant speed, so the lower plot is curvature against distance driven. κ is positive when the road bends left (anticlockwise), negative when it bends right. The evolute runs off to infinity wherever κ passes through 0 (the sine wave’s inflection points, the race track’s chicanes); the cardioid’s cusp, where the speed of the parametrisation drops to zero, is where the curvature blows up.
For a plane curve (x(t), y(t)) the signed curvature is κ = (x′y″ − y′x″)/(x′² + y′²)^(3/2). It is positive when the curve turns left and negative when it turns right, and with respect to arc length it is exactly the rate of change of the heading angle φ: κ = dφ/ds. Driving at a steady speed, the angle of your steering wheel is (to a good approximation for a car) proportional to κ, so the plot underneath is a record of how you steered.
The centres of curvature trace the evolute. Christiaan Huygens studied evolutes in his 1673 Horologium Oscillatorium, while designing pendulum clocks: he showed that the evolute of a cycloid is another cycloid, and used this to make a pendulum whose period does not depend on its swing. Each normal line of the curve is tangent to the evolute, and unwinding a taut string from the evolute traces the original curve, its involute.
The evolute of an ellipse is a stretched astroid with four cusps, one for each vertex, where the curvature reaches a maximum or minimum. Wherever κ passes through 0, at an inflection point, the radius of curvature becomes infinite and the evolute escapes to infinity. Road and railway engineers join straight sections to circular arcs with clothoids, curves whose curvature grows in proportion to arc length, so that the steering wheel turns smoothly instead of jerking.
A planet sweeps out a new shaded sector in every equal stretch of time. Change the eccentricity and compare the measured areas.
Units: AU and years, with GM = 4π² AU³/yr², so every orbit with a = 1 AU takes exactly one year, whatever its eccentricity (Kepler’s third law). Positions come from solving Kepler’s equation, and the sector areas are measured from the drawn polygons, not from the law. The velocity arrow is to scale; the acceleration arrow’s length grows like √|a| so that it stays visible far from the sun, but its direction and its split into a_T and a_N are exact.
Johannes Kepler published his first two laws in 1609, in Astronomia Nova: planets move on ellipses with the sun at a focus, and the line from the sun to a planet sweeps out equal areas in equal times. Newton explained both in the Principia (1687) from an inverse-square force aimed at the sun.
The second law needs only the direction of the force. In a short time dt the planet moves by v dt, and the thin triangle swept has area dA = ½|r × v dt|, so dA/dt = ½|r × v|. Now differentiate: d/dt (r × v) = v × v + r × a. The first term is zero because any vector crossed with itself is zero, and the second is zero because a points along r. So r × v is constant, the angular momentum per unit mass, and area is swept at the constant rate ½|r × v|. Since r × v also keeps its direction, the orbit stays in one plane.
Acceleration always splits along the path as a = (d|v|/dt) T + κ|v|² N. The tangential part a_T changes the speed and the normal part a_N bends the path. On a Kepler orbit the pull of the sun has a tangential part everywhere except at perihelion and aphelion, which is why the planet speeds up as it falls in and slows as it climbs out. For an ellipse with semi-axes a and b the total area is πab, and the period, 2π√(a³/GM), depends only on a: Kepler’s third law.
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