This page is a sub-page of the page on our Mathematical Explainatorium.
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Related KMR-pages:
• Art
• Arcimboldo
• Oscar Reutersvärd
• M.C. Escher
• Surrealism
• Shift of Basis in Art
• Conformal Face Mapping
• The Moebius Strip
• The Klein Bottle
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Related sources of information:
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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.
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Escher’s Angels and Devils moving in the Riemann-Poincaré disk:
• The Riemann-Poincaré model
• The Beltrami-Klein model
• Hyperbolic Geometry
• Non-Euclidean Geometry
• Metric Geometry
• Projective Geometry
• Projective Metrics
• The Euclidean Degeneration
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“Heaven and Hell” in hyperbolic space:
(Vladimir Bulatov on YouTube)
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The Feminine Archetypical Form(ula): \, z \, = \, \nabla \cdot \dfrac{\mathbf{V}}{| \, \mathbf{V} \, |} \; , \; where \; \mathbf{V} = \begin{bmatrix} y \\ x \\ z \end{bmatrix} .
• Vector Analysis
• Vektoranalys (in Swedish)
• Gradients
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Devil transformed by the complex exponential function: \, w \, = \, e^{z + \textcolor{red}{p}} \, :
Devil transformed by complex sin: \, w \, = \, \sin(z + \textcolor{red}{p}) \, :
• Conformal Mappings (Ambjörn Naeve on YouTube)
• Conformal Face Mapping
• Conformal Mapping
• Geometric Numbers
• Complex Numbers
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Complex Multiplication:
Complex Spiralization:
(this video was created by mistake, while trying to simulate complex multiplication)
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The Radian Angle Measure:
Explaining the Radian Angle Measure:
Interactive simulation of the Radian angle measure.
Trigonometry:
The Sin and Cos functions:
Interactive simulation of the Sin and Cos functions.
Interactive simulation of the Sin and Cos functions extended.
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A Trigonometric Inequality:
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• Trigonometry
• Complex Trigonometry
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Stereographic projection of a spherical circle:
[Stereographic projection means projection from the north pole of the sphere
onto the tangent plane at its south pole]:
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• Stereographic Projection
• The Mercator Map Projection
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Möbius Transformations Revealed
(Jonathan Rogness on YouTube):
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Reflecting a plane wavefront in a saddle surface turns it into a Dupin Cyclide:
• Dupin Cyclides
• Dupin Cyclides as Inversions of Circular Cones
• Canal Surfaces
• Focal Surfaces
• Geometric Optics
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Parallel wave fronts of a hyperbolic paraboloid:
• Focal sheets
• Conics
• Optical Properties of Conics
• Quadrics
• Optical Properties of Quadrics
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Double-Cylindrical Point Focusing Mirrors
A practical realization of the DC-Point Focus:
(by Thomas Elofsson, 1989):
• Pointfocus 3D
• Pointfocus 2D
• The Double-Cylindrical Point Focus
Talking heads:
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Confocal Quadrics:
Confocal Quadrics 2 – – Elliptic Coordinates – Intersect XY-plane:
The interactive simulation that created this movie.
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The Tangent Developable of a Circular Helix:
The interactive simulation that created this movie.
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Inversion of a Tangent Surface (with its Principal Net):
Inversion of a Tangent Surface (with its Osculating Planes):
• Canal surfaces
• Inversion
• Inversive Geometry
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The Rectifying Developable of a Circular Helix:
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Isometric Deformation: from Catenoid to Helicoid and back again:
• Isometric Deformations
• Differential Geometry
• Geodesic Curves
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Parametric Surface – approximate Weingarten Map :
The interactive simulation that created this movie.
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Circular helix – square profile Monge surface:
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PSTC surface:
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CircularHelix-Ellipse-PSTC.guts (n = 2.2):
CircularHelix-Ellipse-PSTC.guts (movie):
CircularHelix-Ellipse-PSTC.guts (movie 2):
/////// OLD:
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Trig fractals (Book 2):
(Dave Gates on YouTube)
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Tesseract = 4-Dimensional Cube:
(throughthedoors on YouTube):
Triact = 3-Dimensional Cube:
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The complex exponential function
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A planar electromagnetic wave:
The interactive simulation that created this movie.
The electric part of the wave: \, E(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, = \, e^{ \, i \,(\mathbf{\hat{k}} \cdot \mathbf{x} \, - \, \omega \, t)} \,
The magnetic part of the wave: \, B(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, = \, e^{ \, i \, (\mathbf{\hat{k}} \cdot \mathbf{x} \, - \, (\omega \, + \, \pi/2) \, t)} \,
The entire wave: \, E_m(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, = \, E(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, + \, B(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \,
Its Poynting vector : \, S \, = \, \frac{1}{{\mu}_0} \, E \, \times \, B
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• Maxwell and Dirac theories as an already unified theory
Conceptual background:
Historical background:
• The Evolution Of Geometric Arithmetic
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The interactive simulation that created this movie.
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Projective pencil-to-pencil transformation:
Dual configuration:
Projective range-to-range transformation:
• Projective Geometry
• Projective Metrics
• The Euclidean Degeneration
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Affine approximation of a function from R^2 to R:
• Differentiation and Affine Approximation in one real variable.
• Differentiation and Affine Approximation in several real variables.
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Plane Curves:
The Evolute of an Ellipse:
• Evolutes and Involutes
• Plane Curves
• Optical Properties of Conics
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The interactive simulation that created this movie.
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Cycloid and Trochoid – orbits:
The interactive simulation that generated this movie.
Drag the blue point at the bottom horizontally to change the Trochoid.
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The interactive simulation that created this movie.
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CycloidEvolute – Involutes:
/////////////////////// THE MOVIE IS MISSING
The interactive simulation that will create this movie.
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Inversion of a polar grid:
The interactive simulation that created this movie.
Interactive simulation of 1/z of a polar grid.
This is very similar to inversion, which would be 1/Conjugate(z).
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Devil-inversion-xy.anim (or, rather, Devil transformed by 1/z):
The interactive simulation that created this movie.
Drag the devil to study the dynamics of the transformation. Note the difference from the video clip. The interactive simulation is a true inversion.
Compare: Interactive inversion of the same Devil in the unit circle.
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EpiTrochoid-center-inverse:
The interactive simulation that created this movie.
/////// Positive pedals
The interactive simulation that created this movie.
/////// Negative pedals
Circle-internal-point-negative-pedal:
The interactive simulation that created this movie.
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Cissoid – from a circle, a free point, and a free line:
The distance between the green point and the blue point (on the black line) is equal to
the distance between the red point and the black point.
The interactive simulation that created this movie.
• Cissoids
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A generic Conchoid:
The interactive simulation that generated this movie.
The green point and the black point can be moved.
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Strophoid – from a circle and two generic points:
The interactive simulation that created this movie.
The green points can be moved.
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Ellipse – Focal Ray Reflect:
The interactive simulation that created this movie.
Ellipse – Focal Wave Reflect:
The interactive simulation that created this movie.
Ellipse – Focal Wave/Ray Reflect:
The interactive simulation that created this movie.
/////// Caustic Curves:
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Ellipse – near-focus caustic:
The interactive simulation that created this movie.
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Ellipse-Vertex-Caustic:
The interactive simulation that created this movie.
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Semi-circle parallel light rays real caustic:
The interactive simulation that created this movie.
Semi-circle parallel light rays virtual caustic:
The interactive simulation that created this movie.
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Taylor Expansion
The Taylor polynomial of degree 1 for the function f(x, y) at the point (a, b):
(also known as The Affine Approximation):
{T_{aylor}^1 (f)}_{(a,b)} = f(a,b) + \frac{\partial f}{\partial x}_{(a,b)} (x-a) + \frac{\partial f}{\partial y}_{(a,b)} (y-b)
The interactive simulation that created this movie.
Den interaktiva simulering som skapade denna film.
Interactive simulation of Taylor expansion of order 1.
Drag the red point in the left window to move the point of expansion.
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The Taylor polynomial of degree 2 for the function f(x, y) at the point (a, b):
{T_{aylor}^2(f)}_{(a,b)} = {T_{aylor}^1 (f)}_{(a,b)} + \dfrac{1}{2} (\frac{\partial^2 f}{\partial x^2}_{(a,b)} (x-a)^2 + 2 \frac{\partial^2 f}{\partial x \partial y}_{(a,b)} (x-a)(y-b) + \frac{\partial^2 f}{\partial y^2}_{(a,b)} (y-b)^2)
The interactive simulation that created this movie.
Den interaktiva simulering som skapade denna film.
Interactive simulation of Taylor expansion of order 2.
Drag the red point in the left window to move the point of expansion.
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The Taylor polynomial of degree 3 for the function f(x, y) at the point (a, b):
{T_{aylor}^3(f)}_{(a,b)} = {T_{aylor}^2 (f)}_{(a,b)} + \dfrac{1}{6} (\frac{\partial^3 f}{\partial x^3}_{(a,b)} (x-a)^3 + 3 \frac{\partial^3 f}{\partial x^2 \partial y}_{(a,b)} (x-a)^2 (y-b) + 3 \frac{\partial^3 f}{\partial x \partial y^2}_{(a,b)} (x-a)(y-b)^2 + \frac{\partial^3 f}{\partial y^3}_{(a,b)} (y-b)^3)
The interactive simulation that created this movie.
Den interaktiva simulering som skapade denna film.
Interactive simulation of Taylor expansion of order 3.
Drag the red point in the left window to move the point of expansion.
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The Taylor polynomial of degree 4 for the function f(x, y) at the point (a, b):
The interactive simulation that created this movie.
Den interaktiva simulering som skapade denna film.
Interactive simulation of Taylor expansion of order 4.
Drag the red point in the left window to move the point of expansion.
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The Taylor polynomials of degrees 1 and 2 for the function f(x, y) at the point (a, b):
The interactive simulation that created this movie.
Den interaktiva simulering som skapade denna film.
Gammal:Interactive simulation of Taylor expansion of orders 1 and 2.
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The Taylor polynomials of degrees 1 and 3 for the function f(x, y) at the point (a, b):
The interactive simulation that created this movie.
Den interaktiva simulering som skapade denna film.
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The Taylor polynomials of degrees 1 and 4 for the function f(x, y) at the point (a, b):
The interactive simulation that created this movie.
Den interaktiva simulering som skapade denna film.
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The Taylor polynomials of degrees 1, 2, 3 for the function f(x, y) at the point (a, b):
The interactive simulation that created this movie.
Den interaktiva simulering som skapade denna film.
Interactive simulation of Taylor expansion of orders 1, 2 and 3.
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The Taylor polynomials of degrees 1, 2, 3, 4 for the function f(x, y) at the point (a, b):
The interactive simulation that created this movie.
Den interaktiva simulering som skapade denna film.
Interactive simulation of Taylor expansions of orders 1, 2, 3, and 4.
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