So a circle is like a line where there's a point "at infinity" connecting the "ends". The stereographic projection takes a 1D circle (living in 2D space) minus the top point to a line. Most approaches give you the idea of the shape, but don't preserve lengths or anything geometric like that. So there's a northern hyper-hemisphere and a southern hyper-hemisphere (both 3d bulks with a 2-sphere boundary) glued along their two-sphere boundary (the "equator", which we said before is a 2-sphere). Similarly, a 3-sphere is just two (curved) balls glued along their boundary. Thus, topologically, the 2-sphere is just two disks glued along their boundary. But the hemispheres are just curved disks. We can imagine cutting the 2-sphere along the equator and being left with two hemispheres. However, in the comments achille hui suggested a nice alternative idea that might be a bit cryptic without further explanation. The people on the plane will see a disk expand and contract.Īs a trick for imagining the 3-sphere, you can do much the same thing as above. This is like a 3-ball moving through a flat plane. If you like, you can imagine that the 4-ball is moving through our 3-space and we can only see the intersection.
#DESMOS 4D SPHERE FULL#
Just like a sphere has disk cross-sections that start as points, get to a maximum size at the equator, and then taper off again, a 4-ball will have 3-ball cross-sections that start as a point, grow to the full radius at the "equator" (which is a 2-sphere) and then taper off. One is to imagine projections and cross sections. Thus you are referring to either the $4$-ball or the $3$-sphere. More specifically the surface of a ball is the 2-sphere (note it's a 2d object) and the whole ball is the 3-ball. "Spherical Coordinates."įrom MathWorld-A Wolfram Web Resource.First, as a commenter mentioned, as a technical term, "sphere" refers to the surface of a ball, not its bulk. Referenced on Wolfram|Alpha Spherical Coordinates Cite this as: Standard Mathematical Tables and Formulae. "Tensor Calculations on Computer: Appendix." Comm. Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Orlando, FL: Academic Press, pp. 102-111, "Spherical Polar Coordinates." §2.5 in Mathematical
To Differential Equations and Probability. Apostol,Ģnd ed., Vol. 2: Multi-Variable Calculus and Linear Algebra, with Applications Spherical coordinates of vector (1, 2, 3) Extreme care is therefore needed when consulting the literature. The following table summarizes a number of conventions The symbol is sometimes also used in place of, instead of, and and instead of. Typically means (radial, azimuthal, polar) to a mathematician but (radial, polar,Īzimuthal) to a physicist. This is especially confusing since the identical Unfortunately, the convention in which the symbols and are reversed (both in meaning and in order listed) is alsoįrequently used, especially in physics. Used in the physics literature is retained (resulting, it is hoped, in a bit lessĬonfusion than a foolish rigorous consistency might engender). Is in spherical harmonics, where the convention The sole exception to this convention in this work Remaining the angle in the - plane and becoming the angle out of that Note that this definition provides a logicalĮxtension of the usual polar coordinates notation, In this work, following the mathematics convention, the symbols for the radial, azimuth, and zenith angleĬoordinates are taken as, , and, respectively. This is the convention commonly used in mathematics. To be distance ( radius) from a point to the origin. Is the latitude) from the positive z-axis Known as the zenith angle and colatitude, When referred to as the longitude), to be the polar angle (also Define to be the azimuthal angle in the - plane from the x-axis That are natural for describing positions on a sphere Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates
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