Spherical orthogonal coordinate system agrees with

Spherical orthogonal coordinate system
Morio Kikuchi
Abstract:
Spherical orthogonal coordinate system agrees with plane orthogonal coordinate system in
coordinates, length, and angle of an intersection. Using spherical orthogonal coordinate system, we
can realize complex sphere to which complex number is indicated with no stereographic projection.
By the coordinate transformation of the inversion which is characterized by swap of origin and point
at inﬁnity, three-dimensional orthogonal coordinates are transformed into new coordinates, namely
three-dimensional spherical orthogonal coordinstes, however coordinates and so forth are constant.
1. Swap of origin and point at inﬁnity
A point in three-dimensional orthgonal coordinate system has coordinates by crossing of 3 planes.
Here, we consider swap of origin and point at inﬁnity. That is, origin and point at inﬁnity in
orthogonal coordinate system become point at inﬁnity and origin respectively. We call such
coordinate system new coordinate system. A point in new coordinate system has coordinates by
crossing of 3 spheres which pass through origin of orthgonal coordinate system. We can connect a
point on a plane in orthgonal coordinate system with a point on a sphere in new coordinate system
by the inversion on a sphere of which centre is origin in orthgonal coordinate system. This is
stereographic projection as we know, however, pole is origin in orthgonal coordinate system
diﬀerently from its general ﬁgures.
2. Spherical orthogonal coordinate system
We explain coordinates on a sphere in new coordinate system. We obtain a circle on a sphere in
new coordinate system which passes through origin in orthgonal coordinate system, namely point at
inﬁnity in new coordinate system as a consequence of stereographic projection of a line on a plane in
orthogonal coordinate system. We call it round line in the sense that it is a line on a sphere in new
coordinate system. A length on a round line which passes through the top of the sphere from the
top of the round line is deﬁned as
K tan θ
In the equation, θ is an angle of of a line segment which passes through a point on the round line
and point at inﬁnity in new coordinate system and a line segment which passes through the top of
the round line and point at inﬁnity in new coordinate system, and K is a constant. K is not
curvature but has only a sense that it is a capital letter K which corresponds to a small letter k.
The equation can be derived from a geometric formula. x coordinate on a sphere in new coordinate
system is deﬁned as
x = K tan θx
In the equation, θx is an angle of a line segment which passes through a point on a round line which
passes through the top of the sphere and point at inﬁnity in new coordinate system and a line
segment which passes through the top of the sphere and point at inﬁnity in new coordinate system,
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and K is the constant. y coordinate on the sphere in new coordinate system is deﬁned as, likewise in
a round line which passes through the top of the sphere and intersects at right angle with the round
line for x coordinate,
y = K tan θy
Coordinates of an intersection of a round line which passes through a point on the round line for x
coordinate and intersects at right angles with that and a round line which passes through a point on
the round line for y coordinate and intersects at right angles with that are (x, y).
We can make such coordinates of a point on a sphere in new coordinate system equal to
orthogonal coordinates of a point on a plane which is in the relation of stereographic projection. For
example, we set a plane z = K and a sphere in new coordinate system in the plus direction of z axis.
Assuming (x, y) to be orthogonal coordinates, coordinates on a sphere are
K tan θx = K(x/K) = x
K tan θy = K(y/K) = y
That is, a point on a sphere and a point on a plane which are in the relation of stereographic
projection agree in coordinates.
For the purpose of agreement of length on a plane and length on a sphere, we make coeﬃcient of
tan θ variable.
(K/ cos φ) tan θ
In the equation, φ is an angle of inclination of a round line from the top of a sphere. Assuming a
length of a line segment which passes through two points on a plane to be s, with the other point on
a plane corresponding to the top of a round line, a length on a sphere is
(K/ cos φ) tan θ = (K/ cos φ){s/(K/ cos φ)} = s
For the reason that stereographic projection is the inversion on a sphere of which centre is origin
in orthogonal coordinate system, an angle of an intersection of two lines on a plane and an angle of
an intersection of two round lines on a sphere agree. Because of agreement of coordinates, length,
and angle of an intersection, we should be able to conclude that coordinates on a sphere are
equivalent to coordinates on a plane. We call system of coordinates on a sphere spherical orthogonal
coordinate system.
The above, namely good correspondence between plane orthogonal coordinates and spherical
orthogonal coordinates can produce agreements of coordinates and so forth on the coordinate
transformation of the inversion in three dimensions.
3. Complex sphere
Complex plane is a plane used for indication of complex number, and complex number is
indicated by orthogonal coordinate system in it. On the other, complex sphere is, so far as I know, a
gathering of points corresponding to points on a complex plane, and two-dimensional coordinates
seem to be not very argued in it. As spherical orthogonal coordinate system agrees with plane
orthogonal coordinate system in coordinates, length, and angle of an intersection, we should be able
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to make it a means to indicate complex number directly on a sphere, shouldn’t we?
References:
[1] Morio Kikuchi, ”On coordinate systems by use of spheres (the 1st, . . . , the 4th)”
(2009)(in Japanese)
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球面直角座標
菊池盛雄
アブストラクト:
球面直角座標は平面直角座標と座標、長さ、交角が一致します。球面直角座標を用いると立体射影
によらずに複素数を表示する複素球面が実現されます。原点と無限遠点のスワップを特徴とする反転
によって 3 次元直角座標を新座標すなわち 3 次元球面直角座標に座標変換しても座標等は不変です。
1. 原点と無限遠点のスワップ
3 次元直角座標における 1 点は三つの平面の交点としてその座標を指定します。ここで、原点と無
限遠点をスワップすることを考えてみます。直角座標の原点が無限遠点、直角座標の無限遠方が原点
となる訳です。このような座標系を新座標と称することにします。新座標では、三つの平面ではなく
直角座標の原点を通る三つの球面で座標を指定します。直角座標の平面上の点と新座標の球面上の点
とは、原点を中心とする球面に関する反転で関係付けることができます。よく知られているように、
これは立体射影です。ただし、一般的な立体射影の図と異なり、極は直角座標の原点となっています。
2. 球面直角座標
新座標の球面上の座標について述べます。直角座標の平面における直線を立体射影すると、新座標
の球面上には直角座標の原点、すなわち新座標の無限遠点を通る円が得られます。これを新座標の球
面上の直線という意味で円直線と称します。球面の頂点を通る円直線上の円直線の頂点からの長さ
は、円直線上の点と新座標の無限遠点とを結ぶ線分が、円直線の頂点と新座標の無限遠点とを結ぶ線
分となす角を θ、K を定数として
K tan θ
と定義します。K は曲率ではなく、小文字 k と対応している大文字 K という意味しかありません。
この式は幾何学のある公式から導出することができます。新座標の球面上の x 座標は、球面の頂点を
通る円直線上の点と新座標の無限遠点とを結ぶ線分が、球面の頂点と新座標の無限遠点とを結ぶ線分
となす角を θx 、K を定数として
x = K tan θx
新座標の球面上の y 座標は、球面の頂点を通り直交する円直線において、同様に
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y = K tan θy
これらの円直線上の x 座標、y 座標の点を直交して通る二つの円直線の交点の座標が (x, y) となり
ます。
このように新座標の球面上の点の座標を定めた訳ですが、これは立体射影の関係にある平面上の点
の直角座標と一致させることができます。たとえば、平面が z = K 、新座標の球面が z 軸の正の向き
とします。平面上の点の直角座標を (x, y) とすれば、球面上の座標は
K tan θx = K(x/K) = x
K tan θy = K(y/K) = y
すなわち、立体射影の関係にある球面上の点と平面上の点は座標が一致します。
平面上の長さと球面上の長さを一致させるには、円直線が球面の頂点を通らない、すなわち円直線
が φ 傾いている場合の円直線上の長さを
(K/ cos φ) tan θ
とします。平面上の 2 点を結ぶ線分の長さを s とすれば、球面上の長さは、平面上の一方の点が円直
線の頂点に対応しているとして
(K/ cos φ) tan θ = (K/ cos φ){s/(K/ cos φ)} = s
立体射影は球面に関する反転ですから、平面上の直線の交角と対応する球面上の円直線の交角は一
致します。座標、長さ、交角が一致するのですから、球面上の座標は平面上の直角座標と同等・等価
なものといえるでしょう。この球面上の座標を球面直角座標と称します。
上に示したように平面直角座標と球面直角座標をうまく対応させると、3 次元において直角座標と
新座標は反転という座標変換に関して座標等を一致させることができます。
3. 複素球面
複素平面は複素数を表示するために用いられる平面であり、直角座標で複素数を表示します。一
方、複素球面は、私の知る限りでは、複素平面上の点に対応する点の集まりであり、そこでの 2 次元
座標はあまり議論されていないように見えます。球面直角座標は平面上の直角座標と座標、長さ、交
角が一致するのですから、これを球面上で複素数を直接表示する手段とすることができるのではない
でしょうか。
参考文献:
[1] 菊池盛雄、”球面を用いた座標系について (第 1 回、. . . 、第 4 回)” (2009)
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