First Fundamental Form

Differential Geometry · Axiom Academy

Measuring distances and angles on parametrized surfaces 1. Definition of the First Fundamental Form Let X(u, v) be a parametrization of a surface. For tangent vectors v and w at a point on the surface, the first fundamental form is defined as: The first fundamental form measures how tangent vectors are "stretched" by the parametrization This is simply the dot product of the tangent vectors, measuring their geometric relationship on the surface. The first fundamental form is completely determined by three scalar functions called the coefficients of the first fundamental form. These are computed from the partial derivatives of the parametrization: When F = 0, the coordinate curves are orthogonal. When E = G = 1 and F = 0, we have an orthonormal parametrization. The first fundamental form can be represented as a 2×2 symmetric matrix . For tangent vectors expressed in coordinates as v = v₁X_u + v₂X_v and w = w₁X_u + w₂X_v , we have: This matrix is called the metric tensor and is always symmetric and positive definite. It encodes all the intrinsic geometry of the surface. The determinant EG - F² is always positive and relates to the area element on the surface. The first fundamental form allows us to compute the arc length of a curve on the surface. For a curve α(t) = X(u(t), v(t)) , the arc length element is: The velocity vector is α'(t) = u'(t)X_u + v'(t)X_v , so its length squared is computed using the first fundamental form:

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