Curvature

Curvature

1. What Curvature Measures

Curvature $κ$ (kappa) at $P$ is the rate at which the tangent direction turns as you move along the curve — how sharply the curve bends. The radius of curvature $ρ = 1/κ$ is the radius of the osculating circle (the circle that best kisses the curve at $P$).

where $s$ = arc length from a fixed point, $ψ$ = angle tangent makes with $Ox$ ($tan ψ = dy/dx$). Large $ρ$ → almost straight; small $ρ$ → sharp bend. For a straight line $κ=0, ρ→∞$; for a circle radius $R$, $ρ=R$ everywhere.

2. Cartesian Formula — $y = f(x)$

With $y' = dy/dx, y'' = d²y/dx²$:

If $y''=0$ (point of inflexion or straight segment), $κ=0$. Derivation: $tan ψ = y' ⇒ sec²ψ dψ/dx = y'' ⇒ dψ/ds = (dψ/dx)/(ds/dx)$ and $ds/dx = √(1+y'²)$.

Parametric $x(t), y(t)$ with $x' = dx/dt, y'=dy/dt$:

Polar $r = f(θ)$: with $r' = dr/dθ, r''$:

At the pole where used, often easier to switch to Cartesian.

3. Centre of Curvature and Evolute

Centre $C(α,β)$ of osculating circle lies on the normal at distance $ρ$ from $P$ towards concave side:

Locus of $C$ as $P$ moves is the evolute of the curve (e.g., evolute of parabola $y²=4ax$ is $27a y² =4(x-2a)^3$). Involute is reverse.

4. Worked Examples

(a) Parabola y^2 = 4ax at vertex (0,0):
    Write as x = y^2/4a ⇒ exchange roles. For y = f(x): y = 2√(a x) singular at vertex,
    use x(y): curvature at (0,0) gives ρ = 2a. Check: osculating circle (x-a)^2 + y^2 = a^2 touches vertex.
    At general point (at^2,2at): ρ = 2a(1+t^2)^{3/2}                           ✓

(b) y = x^3 at (1,1): y'=3x^2=3, y''=6x=6 ⇒ ρ=(1+9)^{3/2}/6 =10√10/6=5√10/3 ✓

(c) Circle x^2+y^2=a^2: y'=-x/y, y''=-a^2/y^3 ⇒ (1+y'^2)^{3/2}= (a^3/|y|^3) ⇒ ρ=a ✓

(d) Astroid x = a cos^3 t, y = a sin^3 t at t=π/4:
    x'=-3a cos^2 t sin t, y'=3a sin^2 t cos t, x'' , y'' → numerator (x'^2+y'^2)^{3/2}= (9a^2 cos^4 sin^2+9a^2 sin^4 cos^2)^{3/2} ; |x'y''-y'x''| = (9a^2/2) sin2t → at π/4 ρ = 3a|sin t cos t| =3a/2 ✓

(e) Equiangular spiral r = a e^{θ cotα}: r'=r cotα, r''=r cot^2 α ⇒ ρ = r / sinα = r cosec α ⇒ ρ ∝ r (grows with spiral) ✓

5. Newton's Method (quick alternative for implicit at origin)

If curve passes through origin and tangent is x-axis, expansion gives $ρ = lim y^2/(2x)$ type — sometimes faster than formula. Mention if examiner asks for Newton's method.

6. Exam Points

• Derive and state $ρ = (1+y'²)^{3/2}/|y''|$ — 3-5 marks, almost certain.
• Evaluate $ρ$ for explicit ($y=x^n$), parametric ($x(t),y(t)$), polar ($r=a(1-cosθ)$) at given point — 5-7 marks, very frequent.
• Find centre of curvature / equation of circle of curvature — 5 marks, frequent.
• Find evolute — 5 marks.
• Show $ρ$ for circle = radius, for straight line $∞$ — 3 marks.