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Bitangent — Definition, Formula & Examples

A bitangent is a line that is tangent to a curve at two distinct points simultaneously. At each of these points, the line just touches the curve without crossing it (locally).

Given a plane curve CC, a bitangent line ℓ\ell is a line that is tangent to CC at two distinct points P1P_1 and P2P_2, meaning ℓ\ell meets CC at each PiP_i with contact order at least two. Equivalently, each PiP_i is a point where ℓ\ell touches CC and the curve and line share the same slope.

How It Works

To find a bitangent to a curve y=f(x)y = f(x), you look for two distinct points (a,f(a))(a, f(a)) and (b,f(b))(b, f(b)) where the tangent line is the same line. This requires three conditions: the slopes at the two points must be equal, f′(a)=f′(b)f'(a) = f'(b), and the line through the two points must actually have that slope, so f(b)−f(a)b−a=f′(a)\frac{f(b) - f(a)}{b - a} = f'(a). Solving these simultaneously gives the bitangent, if one exists.

Worked Example

Problem: Find the bitangent line to the curve y=x4−2x2y = x^4 - 2x^2.
Step 1: Compute the derivative to get the slope at any point.
f′(x)=4x3−4xf'(x) = 4x^3 - 4x
Step 2: By symmetry, try tangent points at x=ax = a and x=−ax = -a. Equal slopes require f′(a)=f′(−a)f'(a) = f'(-a). Since f′(−a)=−4a3+4a=−f′(a)f'(-a) = -4a^3 + 4a = -f'(a), the slopes are negatives of each other, so equal slopes forces f′(a)=0f'(a) = 0, giving 4a3−4a=04a^3 - 4a = 0, so a=0a = 0 or a=±1a = \pm 1. Taking a=1a = 1 and b=−1b = -1.
f(1)=1−2=−1,f(−1)=1−2=−1f(1) = 1 - 2 = -1,\quad f(-1) = 1 - 2 = -1
Step 3: Check the slope between the two points: (f(1)−f(−1))/(1−(−1))=0(f(1) - f(-1))/(1 - (-1)) = 0. The slope at x=1x = 1 is f′(1)=4−4=0f'(1) = 4 - 4 = 0. Both conditions match, confirming a bitangent.
y=−1y = -1
Answer: The horizontal line y=−1y = -1 is bitangent to y=x4−2x2y = x^4 - 2x^2, touching the curve at (1,−1)(1, -1) and (−1,−1)(-1, -1).

Why It Matters

Bitangent lines appear in algebraic geometry: a smooth quartic plane curve has exactly 28 bitangent lines, a classical result due to Plücker. They also arise in computational geometry and convex hull algorithms, where external bitangent lines to pairs of convex shapes help determine shared tangent supports.

Common Mistakes

Mistake: Confusing a bitangent with a secant line that merely intersects a curve at two points.
Correction: A secant crosses the curve at two points, while a bitangent must be tangent (matching the curve's slope and touching without locally crossing) at both points.

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