Spherical surface
Spherical surfaceWith sphere. The spherical surface in mathematics (sphere) with from the fixed point of a certain space distance is fixed it is defined. It points to those inside the especially 3 dimensional spaces.
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Summary
In the xyz space, (X0, Y0, Z0) From distanceRPoint (X, Y, Z)
- < Math> (X - X_0) ^2 + (y - y_0) ^2 + (z - z_0) ^2 = r^2 < /math>
It fills up. This system is satisfied, (X, Y, Z) The whole is the spherical surface. EspeciallyRWhen =1 being,Unit spherical surfaceWith you say.
As for area of spherical surface 4 π r2So it is.
Spherical surface plane surface mixes, as for the intersection circle. Especially, it is the center of sphere, (X0, Y0, Z0) It passes plane surface and the spherical surface as for the circle which, among this kind of circles radius largeGreat circle(Is, say it is) with it is called.
earth as sphere, if is, equator and longitude line and the like is example of the great circle. In addition, other than equator latitude line is not the great circle.
N dimensional spherical surface
N As a natural number, the truth N+1 as subclass of the dimensional Euclidean space NDimensionUnit spherical surface SN
- SN = {(X1, ..., XN, XN+1) ∈ RN+1 | (X12 + … + XN2 + XN+12)1/2 = 1}
So it is gathering of the point which is displayed. Not to be the unit spherical surface, radiusR N Dimensional spherical surfaceIf so it is,
- {(X1, ..., XN, XN+1) ∈ RN+1 | (X12 + … + XN2 + XN+12)1/2 = R}
With it should have done.
In addition, whole real number R Whole complex number C Exchanging, be able to define similar ones, the unit spherical surface in that case SN(C) And so on with there are also times when you display. Of course this C Not to be to limit, more other ones (other things metric space and the like) with also it is possible to exchange.
If it is the unit spherical surface of 3 dimensional space which is explained at on, S2 With it means to say.
On the one hand, unit circle spherical surface of 1 dimension S1 So it is and, even if the space R2 So being, not to be Euclid distance 1- norm D1(X, Y) = |X1 - Y1| + |X2 - Y2| (for X = (X1, X2), Y = (Y1, Y2If)) you bring up, perhaps) the unit spherical surface which is defined with this (the one, unit circle is better,
- 4 points (1, 0) and (0, 1), (-1, 0) and (0, -1) the quadrilateral which is made apex {(X, Y) ∈ R2 | |X| + |Y| = 1}
In order so to be, one wind there are times when also those which change call the spherical surface.
Projective straight line
Real number straight line R Point {XXINF} the space which is added P1(R) With writingProjective straight lineWith you say. topology as for this it is circumference. As for this as follows it should have done it is understood.
R X Thinking as the axis Xy- Of plane surface is thought, starting point (0, 0) with the point (0, 1) the circle which designates the straight line which is tied as diameter is drawn there. Furthermore, the point (0, 1) the rectilinear family which it passes Y = Mx When you think of + 1, as for each straight line circle X Because each one mixes with to the axis simply in only point, each point with respect to the circumference which excludes the point (0, 1) X The point on the axis it is possible to make pair correspond one. Then the point (0, 1) hypothetical corresponds infinite-point {XXINF} gathering which is added P1(R) With circumference it becomes in-phase.
Projective plane
(Stub)
In addition
Relation
- sphere
- circle
- unit circle
- metric space
- topology
- Projective geometry
