This page is a sub-page of our page on Metric Geometry.
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The sub-pages of this page are:
• Hyperbolic Geometry
• Elliptic Geometry
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Related pages:
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Books:
• Felix Klein (1926), Vorlesungen über Nichteuklidische Geometrie,
Verlag von Julius Springer in Berlin, (1928).
• Jürgen Richter-Gebert (2011), Perspectives on Projective Geometry – A Guided Tour Through Real and Complex Geometry, Springer, ISBN 978-3-642-17285-4.
• D.M.Y. Sommerville (1914), The Elements of Non-Euclidean Geometry, Dover (1958, 2005).
• Henry Parker Manning (1901), Introductory Non-Euclidean Geometry, Dover (1963, 2005).
• H.S.M. Coxeter (1947 (1942)), Non-Euclidean Geometry.
• W. T. Fishback (1969), Projective and Euclidean Geometry, John Wiley & Sons, Inc.,
ISBN 13: 978-047126-053-0.
• John Stillwell (2016), Elements of Mathematics – From Euclid to Gödel,
Princeton University Press, ISBN 978-0-691-17854-7.
• John Stillwell (1999, (1989)), Mathematics and Its History, Springer, ISBN 0-387-96981-0.
• Jeremy Gray (2007), Worlds Out of Nothing – A Course in the History of Geometry in the 19th Century, Springer, ISBN 1-84628-632-8.
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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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Related Sources of information :
• Projective geometry
• Non-Euclidean Geometry (MathRehab on YouTube)
• Hypercycle Geometry
• Cayley-Klein Metric
• Sommerville, D. M. Y., (1914), The Elements of Non-Euclidean Geometry, Dover, 2005.
/////// Quoting Sommerville (1914/2005, pp. 153-154):
REPRESENTATIONS OF NON-EUCLIDEAN GEOMETRY IN EUCLIDEAN SPACE
1. The problem of representation is one that faces us whenever we try to realise the figures of non-euclidean geometry. There already exists in the mind, whether intuitively or as the result of experiences, a more or less clear idea of euclidean geometry. This geometry has from time immemorial been applied to the space in which we live; and now, when it is pointed out to us that there are other conceivable systems of geometry, each as self-consistent as Euclid‘s, it is a matter of the greatest difficulty to conjure up a picture of space endowed with non-euclidean properties.
The image of euclidean space constantly presents itself and suggests as the easiest solution of the difficulty presented by a representation of non-euclidean geometry by the figures of euclidean geometry.
Thus, upon a sheet of paper, which is for us the rough model of a euclidean plane, we draw figures to represent the entities of non-euclidean geometry. Sometimes we represent the non-euclidean straight lines by straight lines and sometimes by curves, according as the idea of straightness or that of shape happens to be uppermost in the mind.
But we must never forget that the figures that we are constructing are only representations, and that the non-euclidean straight line is every bit as straight as its euclidean counterpart.
The problem of representing non-euclidean geometry on the euclidean plane
is exactly analogous to that of map-projection.
/////// End of quote from Sommerville (1914/2005)
/////// Quoting Sommerville (1914/2005) pp. 125-131:
ANALYTICAL GEOMETRY
1. Coordinates
We shall assume elliptic geometry as the standard case, and construct a system of coordinates. The formulae can be adapted immediately to hyperbolic geometry by changing the sign of \, k^2 \, .
Take two rectangular axes Ox, Oy . Let P be any point, and draw the perpendiculars PM = u and PN = v . Let OP = r and xOP = \theta .
\, r, \theta \, are the polar coordinates of the point P . \,\, u, v \, might be taken as its rectangular coordinates, but we shall find it more convenient to take certain functions of these.
We have
\, \sin \frac{u}{k} = \sin \frac{r}{k} \cos \theta \, ,
\, \sin \frac{v}{k} = \sin \frac{r}{k} \sin \theta \, .
For any point on \, OP, \, therefore, \, \sin \frac{v}{k} = \sin \frac{u}{k} \tan \theta \, .
This is the equation of \, OP, \, in terms of the coordinates \, u \, and \, v \, .
Consider any line. Draw the perpendicular \, ON = p, \, and let \, xON = \alpha \, .
//// Fig.78
\, p \, and \, \alpha \, are always real, and completely determine the line. If \, P \, is any point on the line with coordinates \, (u,v) \,
\, \tan \frac{p}{k} \cot \frac{r}{k} = \cos (\theta - \alpha) \, .
Therefore \, \tan \frac{p}{k} \cos \frac{r}{k} = \sin \frac{u}{k} \cos \alpha + \sin \frac{v}{k} \sin \alpha \, .
This equation is linear and homogeneous in \, \sin \frac{u}{k}, \sin \frac{v}{k}, \cos \frac{r}{k} \, .
We shall effect a great simplification, therefore, if we take as coordinates certain multiples of these functions. The equation of a straight line being now of the first degree, the degree of any homogeneous equation in these coordinates gives the number of points in which a straight line meets the curve, i.e. the degree of the equation is the same as the degree of the curve.
In order that the coordinates of a real point may be real numbers, both in elliptic and in hyperbolic geometry, we shall define the coordinates as follows:
\, x = k \sin \frac{u}{k} = k \sin \frac{r}{k} \cos \theta \, ,
\, y = k \sin \frac{v}{k} = k \sin \frac{r}{k} \sin \theta \, ,
\, z = \cos \frac{r}{k} \, .
These are called Weierstrass’ point-coordinates.
The three homogeneous coordinates are connected by a fixed relationship. We have
\, x^2 + y^2 = k^2 \sin^2 \frac{r}{k} = k^2 (1 - z^2) \, ,
i.e.
\, x^2 + y^2 + k^2 z^2 = k^2 \, .
As any equation in \, x, y, z \, may be made homogeneous by using this identical relation, we need only, in general, use the ratios of the coordinates.
2. The absolute
In hyperbolic geometry, putting \, i k \, instead of \, k \, , we find the coordinates
\, x = k \sinh \frac{u}{k} \, ,
\, y = k \sinh \frac{v}{k} \, ,
\, z = \cosh \frac{r}{k} \, ,
and \, x, y, z \, are connected by the relationship
\, x^2 + y^2 - k^2 z^2 = -k^2 \, .
If \, r \, is infinite, \, x, y, z \, are all infinite, but they have definite limiting ratios. Let \, \alpha, \beta, \gamma \, be the actual values, \, x, y, z \, the ratios, so that \, \alpha = \lambda x, \beta = \lambda y, \gamma = \lambda z \, , and \, \lambda \to \infty \, . Then
\, \alpha^2 + \beta^2 - k^2 \gamma^2 = -k^2 \, ;
therefore
\, x^2 + y^2 - k^2 z^2 = -\frac{k^2}{\lambda^2} = 0 \, .
Hence the ratios of the coordinates of a point at infinity satisfy the equation
\, x^2 + y^2 - k^2 z^2 = 0 \, .
This is the equation of the absolute, which is therefore a curve of the second degree or a conic. In hyperbolic geometry it is a real curve.
In elliptic geometry the equation of the absolute is \, x^2 + y^2 + k^2 z^2 = 0 \, ,
which represents an imaginary conic.
3. Normal form of the equation of a straight line, Line-coordinates
We found the equation of a straight line in terms of the perpendicular \, p \, and the angle \, \alpha \, , which this perpendicular makes with the x-axis, in the form
\, x \cos \alpha + y \sin \alpha = k z \tan \frac{p}{k} \, ,
which may be written
\, \xi x + \eta y + \zeta z = 0 \, .
The ratios \, \xi : \eta : \zeta \, determine the line, and can be taken as its line-coordinates. It is convenient to take certain multiples of these as actual homogeneous coordinates, viz.
\, \xi = \cos \alpha \cos \frac{p}{k} \, ,
\, \eta = \sin \alpha \cos \frac{p}{k} \, ,
\, \zeta = -k \sin \frac{p}{k} \, ,
which are connected by the identical relation
\, k^2 \xi^2 + k^2 \eta^2 + \zeta^2 = k^2 \, .
These are called Weierstrass’ line-coordinates.
\, \xi = \cos \alpha \cosh \frac{p}{k} \, ,
\, \eta = \sin \alpha \cosh \frac{p}{k} \, ,
\, \zeta = -k \sinh \frac{p}{k} \, ,
and the identical relation is
\, k^2 \xi^2 + k^2 \eta^2 - \zeta^2 = k^2 \, .
If \, p \to \infty \, , \, \xi, \eta, \zeta \, \text{all} \to \infty \, .
Let the actual values be \, \alpha, \beta, \gamma \, ,
and let \, \alpha = \lambda \xi, \beta = \lambda \eta, \gamma = \lambda \zeta \, ; then
\, k^2 (\xi^2 + \eta^2) - \zeta^2 = \frac{k^2}{\lambda^2} = 0 \, .
Hence the coordinates of a line at infinity satisfy the equation
\, k^2 \xi^2 + k^2 \eta^2 - \zeta^2 = 0 \, .
A homogeneous equation in line-coordinates \, \xi, \eta, \zeta \, represents an envelope of lines. This equation represents an envelope of class 2, i.e. a conic. This is the same conic as we had before and represents the absolute, since it expresses the condition that the line \, (\xi, \eta, \zeta) \, should be a tangent to \, x^2 + y^2 - k^2 z^2 = 0 \, .
4. Distance between two points
Let \, P(x, y, z) \, \text{and} \, P'(x', y', z') \, be the two points, \, PP' = d \, .
Then, if the polar coordinates are \, (r, \theta) \, and \, (r', \theta') \, ,
\, \cos \frac{d}{k} = \cos \frac{r}{k} \cos \frac{r'}{k} + \sin \frac{r}{k} \sin \frac{r'}{k} \cos (\theta - \theta') = z z' + \frac{x x'}{k^2} + \frac{y y'}{k^2} \, ,
or, in terms of the ratios of the coordinates
\, \cos \frac{d}{k} = \frac{xx'+yy'+k^2zz'}{\sqrt{x^2+y^2+k^2z^2} \sqrt{x'^2+y'^2+k^2z'^2}} \, .
It is convenient to introduce here a brief notation.
If \, (x, y, z), (x', y', z') \, are the coordinates of the two points, we shall define
\, xx'+yy'+k^2zz' \equiv (xx') \, ,
and we shall speak of the points \, (x) \, and \, (x') \, .
Then the distance between the points \, (x) \, and \, (x') \, is given by
\, \cos \frac{d}{k} = \frac{(xx')}{\sqrt{(xx)} \sqrt{(x'x')}} \, .
5. In elliptic geometry the distance-function is periodic
Suppose \, d = \frac{1}{2} \pi k \, ; then \, \cos \frac{d}{k} = 0 \, , and
\, xx'+yy'+k^2zz' = 0 \, ,
i.e. all points on this line are at the distance \, \frac{1}{2} \pi k \, or a quadrant from \, (x', y', z') \, .
This is therefore the equation of the absolute polar of \, (x', y', z') \, .
It is the polar with respect to the conic \, x^2 + y^2 + k^2 z^2 = 0 \, .
This is therefore the equation of the absolute.
=======
Suppose \, d = \pi k \, ; then \, \cos \frac{d}{k} = -1 \, , and, with actual values of the coordinates,
\, xx'+yy'+k^2zz'=-k^2 \, ,
but \,\,\,\, x^2 + y^2 + k^2 z^2 = k^2 \, ,
and \,\, x'^2 + y'^2 + k^2 z'^2 = k^2 \, ;
therefore, multiplying the first equation by 2 and adding to the others,
\, (x+x')^2 + (y+y')^2 + k^2 (z+z')^2 = 0 \, ,
which requires that \, x' = -x, y' = -y, z' = -z \, .
In spherical geometry these would represent antipodal points. In elliptic geometry antipodal points coincide, and therefore in every case, if two points have their coordinates in the same ratios, they must coincide.
6. Angle between two lines
//// Dually one can derive (see Sommerville (1914/2005, pp. 131-132)
If \, (\xi \xi) = 0 \, is the line-equation of the absolute,
\, \cos \phi = \frac{(\xi_1 \xi_2)}{\sqrt{(\xi_1 \xi_1)} \sqrt{(\xi_2 \xi_2)}} \, .
/////// End of quote from Sommerville (1914/2005)
/////// Quoting Sommerville (1914/2005, pp. 136-137):
12. The circle
A circle is the locus of points equidistant from a fixed point. Let \, (x_1, y_1, z_1) \, be the centre and \, r \, the radius; then the equation of the circle is
\, \cos \frac{r}{k} = \frac{xx_1+yy_1+zz_1}{\sqrt{x^2+y^2+k^2z^2} \sqrt{{x_1}^2+{y_1}^2+k^2{z_1}^2}} \, ,
or, when rationalised,
\, (x x) (x_1 x_1) \cos^2 \frac{r}{k} = (x x_1)^2 \, .
This equation is of the second degree, and from its form we see that it represents a conic touching the absolute \, (x x) = 0 \, at the points where it is cut by the line \, (x x_1) = 0 \, .
\, (x x_1) = 0 \, is the polar of the centre, and is therefore equidistant from the circle,
i.e. it is the axis of the circle.
Hence A circle is a conic having double contact with the absolute ; its axis is the common chord and its centre is the pole of the common chord.
The equidistant-curve
Let \, (\xi_1, \eta_1, \zeta_1) \, be the coordinates of the axis, and \, d \, the constant distance; then the equation of the curve is
\, \sin \frac{d}{k} = \frac{\xi_1 x + \eta_1 y + \zeta_1 z}{\sqrt{{\xi_1}^2 + {\eta_1}^2 + {\zeta_1}^2/k^2} \sqrt{x^2 + y^2 + k^2 z^2}} \, ,
or
\, (x x) (\xi \xi) \sin^2 \frac{d}{k} = (\xi_1 x + \eta_1 y + \zeta_1 z)^2 \, .
This again represents a conic having double contact with the absolute, the common chord being the axis. The pole of the axis is equidistant from the curve, and so the equidistant-curve is a circle. In elliptic geometry both centre and axis are real, in hyperbolic geometry the centre alone is real for a proper circle, and the axis alone is real for an equidistant-curve.
The horocycle
In hyperbolic geometry, the equation of the absolute being \, x^2 + y^2 - k^2 z^2 = 0 \, ,
the equation of a horocycle is of the form
\, x^2 + y^2 - k^2 z^2 = \lambda (a x + b y + c z)^2 \, ,
where \,a^2 + b^2 = \frac{c^2}{k^2} \, .
/////// End of quote from Sommerville (1914/2005)
Concentric ordinary circles in hyperbolic geometry:
Concentric horocycles in hyperbolic geometry:
Concentric hypercycles in hyperbolic geometry:
/////// Quoting Sommerville (1914/2005, pp. 140-141):
16. Explanation of apparent exception in euclidean geometry
[…]
The conception of a pair of parallel straight lines as forming a circle in euclidean geometry is consistent with the definition of a circle as a conic having double contact with the absolute, for the absolute in this case is a pair of coincident straight lines, and this is cut by a pair of parallel lines in two pairs of coincident points. A single straight line is not, of course, a tangent to the absolute, though it cuts it in two coincident points ; this case is just the same as that of a line which passes through a double point on a curve, but which is not considered as being a tangent.
But when we have a pair of parallel lines cutting the absolute \, \Omega \, in four points all coincident, we can regard \, \Omega \, as being a tangent to the curve consisting of this pair of lines. Fig 80 represents the case approximately when the absolute is still a proper conic and the pair of straight lines is also a proper conic, having double contact with the absolute
///// Fig 80
The axis of the circle consisting of a pair of parallel lines is the line lying midway between them ; the absolute pole of this (a point at infinity) is the centre.
When the axis passes through the centre, i.e. when it coincides with the line at infinity, the circle becomes a horocycle, which is thus represented in euclidean geometry by a straight line together with the line at infinity.
Two equidistant-curves, with parallel axes, have the same centre at infinity. In hyperbolic geometry two equidistant-curves, with parallel axes intersecting at infinity at \, O \, , have their centres on the tangent at \, O \, , and therefore at a zero distance apart though not coincident.
/////// FORTSÄTT HÄR
/////// End of quote from Sommerville (1914/2005)
/////// Quoting Sommerville (1914/2005, pp. 154-156):
Projective Representation
2. The fact that a straight line can be represented by an equation of the first degree enables us to represent non-euclidean straight lines by straight lines on the euclidean plane. Distances and angles will not, however, be truly represented, and we must find functions of the euclidean distances and angles which give the actual distances and angles of non-euclidean geometry.
3. The absolute is represented by a conic. In hyperbolic geometry this conic is real, in elliptic geometry it is wholly imaginary, but in every case the polar of a real point is a real line. The conic always has a real equation. In the case in which the absolute is a real conic, we could, if we like, represent it by a circle, but except in special cases this does not give any gain in simplicity.
Two lines whose point of intersection is on the absolute are parallel; two lines whose point of intersection lies outside the absolute are non-intersectors. The points outside the absolute have to be regarded as ultra-infinite, and are called ideal points. They are distinguished from other imaginary points by the fact that, while their actual coordinates are all imaginary, the ratios of their coordinates are real. In the present representation they are represented by real points; other imaginary points are represented by imaginary points. (Cf. Chap. IV. §8.)
A real line has two points at infinity, and part of the line lies in the ideal region. A line which touches the absolute has one point at infinity , and all the rest of the line is ideal. A line which lies outside the absolute is wholly ideal.
////// Fig 86 (p. 155)
Through any point two parallels can be drawn to a given line, viz. the lines joining the given point to the two points at infinity on the given line. A triangle which has its three vertices on the absolute has a constant area.
In elliptic geometry the absolute is imaginary, and there are no ideal points.
///////
6. Now, in ordinary geometry
the angle between two lines can be expressed in terms of the two lines joining their point of intersection to the circular points. [Footnote: E. Laguerre, “Note sur la théorie des foyers,” Nouv. Ann. Math., Paris, 12 (1853)]
[…]
i.e. the angle between two lines is a certain multiple of the logarithm of the cross-ratio of the pencil formed by the two lines and the lines joining their point of intersection to the circular points.
7. Now let us return to the case where the absolute is a real conic
\, x^2 + y^2 - k^2 z^2 = 0 \, .
Consider two points \, P(x, y, z), P'(x', y', z') \, .
The point \, (x + \lambda x', y + \lambda y', z + \lambda z') \, lies on their join.
If this point is on the absolute,
\, (x + \lambda x')^2 + (y + \lambda y')^2 - k^2 (z + \lambda z')^2 = 0 \, ,
i.e.
\, \lambda^2 (x'^2 + y'^2 - k^2 z'^2) + 2 \lambda (x x' + y y' - k^2 z z') + (x^2 + y^2 - k^2 z^2) = 0 \, .
Let \, \lambda_1, \lambda_2 \, be the roots of this quadratic. The line \, PP' \, cuts the absolute in the two points \, X, Y \, , corresponding to these parameters, and the cross-ratio of the range
\, (PP', XY) = \frac{\lambda_1}{\lambda_2} \, .
Let \, (PP') = d = k \phi \, , and
\, x^2 + y^2 - k^2 z^2 = r^2 \, , \, x'^2 + y'^2 - k^2 z'^2 = r'^2 \, ;
then the quadratic for \, \lambda \, becomes
\, \lambda^2 r'^2 + 2 \lambda r r' \cos \phi + r^2 = 0 \, ;
whence \, \lambda_1, \lambda_2 = (- \cos \phi \stackrel{+}{-} \sqrt{- \sin^2 \phi}) r/r' = -e^{\stackrel{+}{-} i \phi} r/r' \, .
Therefore \, \lambda_1 / \lambda_2 = -e^{-2 i \phi} \, and \, \phi = \frac{1}{2} i \log (PP', XY) \, .
Therefore, \, d = \frac{1}{2} i k \log (PP', XY) \, ,
i.e. the distance between two points is a certain multiple of the logarithm of the cross-ratio of the range formed by the two points and the two points in which their join cuts the absolute.
In a similar way it can be shown that
the angle between two straight lines is a certain multiple of the logarithm of the cross-ratio of the pencil formed by the two lines and the two tangents from their point of intersection to the absolute.
If the unit angle is such that the angle between two lines which are conjugate with regard to the absolute is \,\frac{1}{2} \pi \, , then
\, \phi = \frac{1}{2} i \log (pp', xy) \, .
8. By this representation the whole of metrical geometry
is reduced to projective geometry,
for cross-ratios are unaltered by projection.
Any projective transformation which leaves the absolute unaltered will therefore leave distances and angles unaltered. Such transformations are called congruent transformations and form the most general motions of rigid bodies.
The projective metric is associated with the name of CAYLEY [Footnote 1] who invented the term Absolute. He was the first to develop the theory of the absolute, though only as a geometrical representation of the algebra of quantics. KLEIN [Footnote 2] introduced the logarithmic expressions and showed the connection between Cayley’s theory and Lobachevsky’s geometry [Footnote 3].
[Footnote 1: “A sixth memoir upon quantics,” London Phil. Trans. R. Soc., 149 (1859). Cayley wrote a number of papers dealing specifically with non-euclidean geometry, but although he must be regarded as one of the epoch-makers, he never quite arrived at a just appreciation of the science. In his mind, non-euclidean geometry scarcely attained to an independent existence, but was always either the geometry upon a certain class of curved surfaces, like spherical geometry, or a mode of representation of certain projective relations in euclidean geometry.]
[Footnote 2: “Über die sogennante Nicht-Euklidische Geometrie,” Math. Ann., 4 (1871), 6 (1873).]
[Footnote 3: Since the definition of the cross-ratio of a range is the same in non-euclidean geometry, the logarithmic expressions for distance and angle hold not only in the euclidean representation of the geometry, but also in the actual non-euclidean geometry itself.]
/////// End of quote from Sommerville (1914/2005)
Conic sections:
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Pappus-Pascal’s theorem:
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Pascal’s theorem:
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Isometry of a non-degenerated geometry:
Hyperbolic circle:
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/////// Quoting Manning (1901/2005 pp. 60-61):
The nature of the angle of parallelism is, therefore, expressed by the equations
\, \sin \Pi(x) = \dfrac{1}{\cos ix} \, ,
\, \tan \Pi(x) = \dfrac{i}{\sin ix} \, ,
\, \cos \Pi(x) = \dfrac{\tan ix}{i} \, .
4. Substituting in the formulæ of plane right triangles, we find that they reduce to those of spherical right triangles with \, ia \, , \, ib \, , and \, ic \, for \, a \, , \, b \, , and \, c \, , respectively. The formulæ of oblique triangles are obtained from those of right triangles in the same way as on the sphere, and thus all the formulae of Plane Trigonometry are obtained from those of Spherical Trigonometry simply by making this change.
As fundamental formulæ for oblique triangles we write
\, \dfrac{\sin A}{\sin i a} = \dfrac{\sin B}{\sin i b} = \dfrac{\sin C}{\sin i c} \, ,
\, \cos i a = \cos i b \cos i c + \sin i b \sin i c \cos A \, ,
\, \cos A = - \cos B \cos C + \sin B \sin C \cos i a \, .
In the notation of the \,\Pi-function, these are
\, \sin A \tan \Pi (a) = \sin B \tan \Pi (b) = \sin C \tan \Pi (c) \, ,
\, \dfrac{\sin \Pi(b) \sin \Pi(c)}{\sin \Pi(a)} = 1 - \cos \Pi(b) \cos \Pi(c) \cos A \, ,
\, \cos A = - \cos B \cos C + \dfrac{\sin B \sin C}{\sin \Pi(a)} \, .
5. Since for very small values of \, x \, we have approximately
\, \sin i x = i x \, ,
\, \cos i x = 1 + \dfrac{x^2}{2} \, ,
\, \tan i x = i x \, ,
our formulæ for infinitesimal triangles reduce to
\, \dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c} \, ,
\, a^2 = b^2 + c^2 - 2 b c \cos A \, ,
\, \cos A = - \cos (B + C) \, .
6. Triangles on an equidistant-surface are similar to their projections on the base plane; that is, they have the same angles and their sides are proportional. Thus the formulæ of Plane Trigonometry hold for any equidistant-surface if with the letters representing the sides we put, besides \, i , a constant factor depending on the distance of the surface from the plane.
/////// End of quote from Manning (1901/2005