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from article Tangent
Line L tangent to curve C at P.
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from article Hyperbola
Hyperbola as a conic section.
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from article Hyperbola
Hyperbola as a locus of points.
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from article Geometrization theorem
Making a 2-D torus.
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from article Geometrization theorem
Life in Seifert-Weber dodecahedral space. (Courtesy of Geometry Center, University of Minnesota)
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from article Geometrization theorem
Hyperbolic dodecahedron of infinite diameter. The infinite spikes are rounded off “chimneys,” which are here triangular rather than rectangular. (Courtesy of Paul Nylander fromhttp://bugman123.com/)
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from article Geometrization theorem
Neck pinching, resulting in infinite curvature.
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from article Geometrization theorem
Ricci flow with surgery.
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from article Matrix theory
Vector rotation that rotates any vector v in the plane counterclockwise through 45° to a new vector Tv.
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from article Calculus of variations
Weierstrass condition. Symbols explained in text.
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from article Calculus of variations
Example of an area-minimizing surface that cannot be written as the graph of a single function. (a) Nonconvex domain C. (b) Minimal graph over C that is not area-minimizing. (c) Second surface over C that has less area. (After F. Morgan, Geometric…
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from article Calculus of variations
Comparison of surfaces. (a) Surface with no handles. (b) Surface with one handle and less area. (c) Surface with two handles and still less area. (d, e, f) These three surfaces are deformed to show that they do have zero, one, and two handles,…
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from article Calculus of variations
Examples of surfaces minimizing the integral of a nonelliptic integrand among all bodies having the same volume (the surface at upper left) and among all surfaces having the same boundary (the saddle-shaped surface). Surfaces were computed as part…
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from article Linear algebra
Geometric representation of a vector OP.
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from article Conic section
Four conic sections. (a) Circle. (b) Ellipse. (c) Parabola. (d) Hyperbola.
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from article Catenary
Catenary and force triangle which keeps a portion of the string in equilibrium.
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from article Catenary
Relation of catenary to the tractrix which is its involute.
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from article Legendre functions
Legendre polynomials Pn(x), n = 2, 3, 4, 5.
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from article Legendre functions
Legendre functions of the second kind, Qn(x). (After E. Jahnke and F. Emde, Tables of Functions with Formulae and Curves, 4th ed., Dover Publications, 1945)
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from article Circle
A circle, with radius, diameter, and circumference indicated. (Credit: McGraw-Hill Education)
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from article Circle
Nine-point circle and Euler line PMH.
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from article Trihedron
Trihedron and trihedral angles.
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from article Trihedron
Spherical triangle ABC formed by intersection of sphere with a trihedron. The center of the sphere is at the vertex of the trihedron.
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from article Gamma function
Depiction of gamma function Γ(z) for real values of z.
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from article Slope
Slope of a line.
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from article Slope
Slope of a curve.
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from article Finite element method
Illustration of a Von Mises stress plot for the bending of an I-beam. The finite element method can be used to investigate these stresses. (Credit: Shutterstock / Mathew Alexander)
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from article Finite element method
Structures modeled by fitting together structural elements: (a) truss structure; (b) two-dimensional planar structure.
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from article Finite element method
Element types: (a) bar; (b) beam; (c) membrane elements; (d) solid elements; (e) axisymmetric solid; (f) plate-bending elements; (g) curved-shell elements; (h) axisymmetric thin-shell element.
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from article Finite element method
Finite element method of (a) fuselage section and (b) wing of a business jet.
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from article Euclidean geometry
Parallel lines and planes.
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from article Euclidean geometry
Normal to a plane.
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from article Euclidean geometry
Regular pentagon ABCDE showing the diagonal d.
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from article Radian measure
Diagrams showing radian measurement. Symbols are explained in the text.
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from article Z transform
Extraction of discrete-time sample values from a continuous signal.
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from article Pythagorean theorem
Proof of Pythagorean theorem based on comparison of areas of squares.
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from article Pythagorean theorem
Three-dimensional Pythagorean theorem.
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from article Analysis of variance
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from article Fourier series and transforms
Examples of (a–c) macrodasyid and (d–f) chaetonotid gastrotrichs. (a) Xenodasys. (b) Dolichodasys. (c) Thaumastoderma. (d) Draculiciteria. (e) Halichaetonotus. (f) Aspidiophorus. (part a after C. Schoepfer-Sterrer, Chordodasys riedli gen. nov.,…
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from article Fourier series and transforms
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from article Fourier series and transforms
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from article Fuzzy sets and systems
Examples of fuzzy sets. (a) Fuzzy sets of age groups of high school children: young (A); not young (A); very young (B). (b) Typical triangular representations of fuzzy labels of temperature: less than 6°C [43°F] (A); about 15°C [59°F] (B); and about…
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from article Fuzzy sets and systems
Dynamic system with IF … THEN rules.
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from article Fuzzy sets and systems
Fuzzy logic controller using IF … THEN control rules. F = fuzzifier; DF = defuzzifier.
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from article Fuzzy sets and systems
Fuzzy control of a direct-current (dc) motor. (a) Elements of motor and controller. ω= motor angular speed, ωZ = reference angular speed, M(t) = disturbing moment, S = error sum, I(t) = motor current, U = control action. (b) Motor speed with fuzzy…
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from article Fuzzy sets and systems
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from article Cube
A cube, a type of parallelepiped.
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from article Differentiation
The natural logarithm function.
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from article Asymptote
Asymptotes of a hyperbola.
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from article Tractrix
Tractrix (upper half).
