Solid surfaces can unfold or unroll so all faces or surfaces lie in one plane {development, solid}.
Surfaces {developable surface} can flatten onto a plane without distortion, so surface line elements become plane line elements.
Continuous transformations of n-simplexes onto themselves have at least one fixed point {fixed point theorem}.
Processes {quadrature} can try to find squares equal in area to surfaces. If plane figure has only straight lines, compass and straightedge can perform quadrature.
Spaces can have simplest geometric figure {simplex, space}| {space, cell}. Number of space dimensions defines simplex: 0 is point, 1 is line, 2 is triangle, 3 is tetrahedron, and n is n-simplex. Simplexes are manifolds. Simplexes have orientation. Even numbers of permutations make same orientation. Odd numbers of permutations make opposite orientation. Simplex boundaries are next-lower-dimension simplexes and have orientation.
Surface areas can use measures {square measure}.
To find area under curve, replace curve with chords, to make equal-width orthogonal projections onto independent-variable axis, and add trapezoid areas {trapezoidal rule}|.
Surfaces {continuous surface} can have tangent plane and normal line at all points.
Truncated solids can have parallel plane sections {frustum}.
Surfaces {hyperboloid} can have cross-sections that are hyperbolas.
If surfaces {isometric surface} bend without stretching and keep one-to-one correspondence, curvature and all other properties stay the same.
Focal surfaces of systems of second-order rays are fourth-degree and class-four surfaces {Kummer surface}. Fresnel wave surfaces are special cases.
Solids can have plane parallel faces {lamina} that are small distance apart compared to face length.
Surfaces {paraboloid} can have sections that are parabolas. x^2 / a^2 + y^2 / b^2 = 2*c*z {elliptic paraboloid}, where a b c are axes. Elliptic paraboloid with a = b is parabola rotated about its z-axis. x^2 / a^2 - y^2 / b^2 = 2*c*z {hyperbolic paraboloid}, where a b c are axes.
Straight lines {rectilinear generator} can generate surfaces {ruled surface}| by translation in one direction, making lines at equal intervals. If there are two distinct generators, surfaces are doubly ruled. No generator can make skew surfaces.
Surfaces {smooth surface} can have no irregularities. Objects on smooth surfaces move only in direction tangent to surface.
Surfaces {unilateral surface} can have one side.
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Date Modified: 2022.0225