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The tangent plane of a regular surface patch
Definition
Let be a regular parametrized surface patch (Regular parametrized surface patches on compact Jordan parameter regions) and let with . At an interior parameter point, the tangent plane is .
The two spanning vectors are linearly independent because their cross product is nonzero (The squared cross-product norm is the Gram determinant of two vectors), so this is a two-dimensional linear subspace of (Linear combination of a finite list, and the span as the smallest linear subspace containing ). The affine tangent plane through is .
Depends on
Used by
- The first fundamental form, Gram matrix, and area density of a surface patch Definition
- Unit normal fields, orientations, and flux through a regular surface patch Definition
- The tangent plane is invariant under regular reparametrization Proposition
- Regular level surfaces have local regular parametrizations with the same tangent plane Theorem
Dependency tree · two levels
20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- M. E. Taylor, Introduction to Analysis in Several Variables, Section 3.2 (standard reference, not scraped)
- University of Toronto MAT237 notes, Section 5.3 (standard reference, not scraped)