The arithmetic mean of a finite nonempty collection of vertices, treating each vertex as an equal point mass. For a polygon, this need not equal the uniform-area centroid of its interior. A translation or linear map commutes with this average; consequently rotating all vertices about their vertex centroid leaves that centroid fixed.
For a planar region of finite positive area , its first moments of area about the coordinate axes are the displayed double integrals, provided they exist. Its centroid is . This follows by applying the definition to each component. The subscript indicates the axis about which the moment is taken: the first moment about the -axis weights by , not .
For a nondegenerate -dimensional simplex with vertices and an affine function , the integral equals the volume times the value at the centroid . To prove this, write in a barycentric coordinate system. On the standard simplex, symmetry permuting the vertices gives equal integrals of all ; since , each is . An invertible affine map transfers this result to any simplex. Affineness gives , so linearity of integration proves the formula. For a tetrahedron the factor is one quarter. The rule is exact for affine integrands; a nonlinear integrand generally needs additional moments.
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A centroid is a fundamental geometric concept referring to the arithmetic center of a shape or a set of points. In different contexts, the term can have specific meanings: 1. **In Geometry**: - The centroid of a simple shape, like a triangle or a polygon, is the point that represents the average position of all the points in the shape. It can be thought of as the center of mass if the shape has uniform density.