Cartographic projections of the sphere

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Problem 2256 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.7 "Space Curves: The Tangent Line and the Normal Plane"

A curve called the loxodrome is defined by the equation $\varphi =a\ln \tg \left(\dfrac {\pi }{4}-\dfrac {\theta }{2}\right)$ , where $\theta$ is the latitude and $\varphi$ is the longitude of a point on a sphere. Prove that it intersects the meridians of the sphere at an angle whose tangent equals $a$ .

Problem 2257 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.7 "Space Curves: The Tangent Line and the Normal Plane"

Prove that stereographic projection gives a conformal image, i.e.that lines in the plane intersect at the same angle as their images on the sphere.

Remark.

Stereographic projection is defined as follows. The topmost point of the sphere $x^{2}+y^{2}+z^{2}=az$ , i.e.the point $A(0,0,a)$ , is joined by a line to an arbitrary point $M(x,y,0)$ of the plane $xOy$ . The point $M_{1}$ , at which the line $AM$ meets the surface of the sphere $x^{2}+y^{2}+z^{2}=az$ , is taken as the image on the sphere of the point $M$ of the plane $xOy$ , and the point $M$ is the image of the point $M_{1}(x_{1},y_{1},z_{1})$ lying on the sphere. Thus stereographic projection gives a one-to-one correspondence between the points of the plane and of the sphere, with the exception of the point $A$ at the top of the sphere. Here

x_{1} = \frac{a^{2}x}{x^{2}+y^{2}+a^{2}}, \qquad y_{1} = \frac{a^{2}y}{x^{2}+y^{2}+a^{2}}, \qquad z_{1} = \frac{a(x^{2}+y^{2})}{x^{2}+y^{2}+a^{2}} .

Problem 2258 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.7 "Space Curves: The Tangent Line and the Normal Plane"

Prove that the curve $r=e^{m\varphi }$ , lying in the plane $xOy$ , where the polar axis coincides with the positive part of the axis $Ox$ , is taken by stereographic projection (see problem 2257) into a loxodrome.

Problem 2259 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.7 "Space Curves: The Tangent Line and the Normal Plane"

Prove that circles on the sphere are taken by stereographic projection (see problem 2257) into circles in the plane, straight lines being regarded as a special case of circles.

Problem 2260 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.7 "Space Curves: The Tangent Line and the Normal Plane"

The central projection of the sphere $x^{2}+y^{2}+z^{2}=2az$ onto the plane $xOy$ is formed as follows. An arbitrary point $M(x,y,0)$ of the plane $xOy$ is joined by a line $AM$ to the center of the sphere $A(0,0,a)$ . The point $M_{1}$ , at which $AM$ meets the sphere, is taken as the image of the point $M$ , and conversely. Prove that this projection is not a conformal image.

References

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.