Spherical trigonometry
Spherical triangles. On the surface of a ball, the shortest distance between two points is measured along the circumference of a great circle, that is, a circle whose plane passes through the center of the ball. Vertices of a spherical triangle are the intersection points of three rays emanating from the center of the ball and the spherical surface. Parties a, b, c A spherical triangle is called those angles between the rays that are smaller (if one of these angles is equal to , then the spherical triangle degenerates into a semicircle of a great circle). Each side of the triangle corresponds to an arc of a great circle on the surface of the ball (see figure).
Angles A, B, C spherical triangle, opposite sides a, b, c accordingly, they are, by definition, angles less than , between arcs of great circles corresponding to the sides of a triangle, or angles between planes defined by these rays.
Spherical trigonometry studies the relationships between the sides and angles of spherical triangles (for example, on the surface of the Earth and on the celestial sphere). However, physicists and engineers prefer to use rotational transformations rather than spherical trigonometry in many problems.
Properties of spherical triangles. Each side and angle of a spherical triangle is by definition smaller.
The geometry on the surface of the ball is non-Euclidean; in every spherical triangle, the sum of the sides is between 0 and , the sum of the angles is between and . In every spherical triangle, the larger angle lies opposite the larger side. The sum of any two sides is greater than the third side, the sum of any two angles is less than plus the third angle.
The story of this demo is this: one day a friend of mine made a planet map generator for his game and wanted the maps created in this way to be shown as a rotating sphere. However, he did not want to use 3D graphics, but instead generated many frames with this very sphere rotated at different angles. The amount of memory used was... let's say, excessive, and the speed of frame generation (as well as the quality of their execution) suffered greatly. After thinking a little, I managed to help him optimize this process, but overall I couldn’t shake the fair feeling that this was a task for OpenGL, and not for 2D graphics at all.
And so, one day, when I was suffering from insomnia, I decided to try to combine these two approaches: draw a rotating sphere (with a planet map stretched over it) through OpenGL, but at the same time leaving it flat.
And I must say that I succeeded. But first things first.
Mathematics of the process
First, let's define the task itself. For each point on the screen, we have two screen coordinates in the Cartesian coordinate system, and we need to find spherical coordinates for it (actually, latitude and longitude), which are essentially texture coordinates for the planet map.
So. The transition from a Cartesian coordinate system to a spherical one is given by a system of equations (taken from Wikipedia):
and the reverse transition - with the following equations:
Coordinate Z we can easily get from X And Y, knowing the radius, and we can take the radius itself equal to one.
In the future, we will agree that we will slightly change the above equations by interchanging the concepts Y(for us this will be the screen vertical) and Z(this will be the depth of the scene).
Technical part
The implementation of the idea will require us to use a quad (I have already written about how to use it, so I will not repeat it, especially since below is a link to the full source code of the project), as well as two textures: the planet map itself (I used an Earth texture of size 2048x1024) and texture coordinate maps. The second texture generation code neatly repeats the mathematics of the conversion from Cartesian to spherical coordinates:
Int texSize = 1024; double r = texSize * 0.5; int pixels = new int; for (int row = 0, idx = 0; row< texSize; row++) { double y = (r - row) / r; double sin_theta = Math.sqrt(1 - y*y); double theta = Math.acos(y); long v = Math.round(255 * theta / Math.PI); for (int col = 0; col < texSize; col++) { double x = (r - col) / r; long u = 0, a = 0; if (x >= -sin_theta && x<= sin_theta) { double z = Math.sqrt(1 - y*y - x*x); double phi = Math.atan2(z, x); u = Math.round(255 * phi / (2 * Math.PI)); a = Math.round(255 * z); } pixels = (int) ((a << 24) + (v << 8) + u); } } GLES20.glGenTextures(1, genbuf, 0); offsetTex = genbuf; if (offsetTex != 0) { GLES20.glBindTexture(GLES20.GL_TEXTURE_2D, offsetTex); GLES20.glTexParameteri(GLES20.GL_TEXTURE_2D, GLES20.GL_TEXTURE_MIN_FILTER, GLES20.GL_NEAREST); GLES20.glTexParameteri(GLES20.GL_TEXTURE_2D, GLES20.GL_TEXTURE_MAG_FILTER, GLES20.GL_NEAREST); GLES20.glTexParameteri(GLES20.GL_TEXTURE_2D, GLES20.GL_TEXTURE_WRAP_S, GLES20.GL_NONE); GLES20.glTexParameteri(GLES20.GL_TEXTURE_2D, GLES20.GL_TEXTURE_WRAP_T, GLES20.GL_NONE); GLES20.glTexImage2D(GLES20.GL_TEXTURE_2D, 0, GLES20.GL_RGBA, texSize, texSize, 0, GLES20.GL_RGBA, GLES20.GL_UNSIGNED_BYTE, IntBuffer.wrap(pixels)); }
Note that the coordinates X And Y are translated from range to range [-1..1], and texture coordinates U And V are converted from radians to the range , after which they are written respectively to the red and green components of the 32-bit texture. The alpha channel is used to preserve "depth" (coordinates Z), and blue remains unused for now. Disabling bilinear filtering is also not accidental: at this stage it does not give any effect (neighboring points in any case have the same values, with rather sharp jumps), and in what I am going to show next, it will harmful. But more on that below.
Private final String quadFS = "precision mediump float;n" + "uniform sampler2D uTexture0;n" + "uniform sampler2D uTexture1;n" + "uniform float uOffset;n" + "varying vec4 TexCoord0;n" + "void main() (n" + " vec4 vTex = texture2D(uTexture0, TexCoord0.xy);n" + " vec3 vOff = vTex.xyz * 255.0;n" + " float hiY = floor(vOff.y / 16.0);n" + " float loY = vOff.y - 16.0 * hiY;n" + " vec2 vCoord = vec2(n" + " (256.0 * loY + vOff.x) / 4095.0 + uOffset,n" + " (vOff.z * 16.0 + hiY ) / 4095.0);n" + " vec3 vCol = texture2D(uTexture1, vCoord).rgb;n" + " gl_FragColor = vec4(vCol * vTex.w, (vTex.w > 0.0 ? 1.0: 0.0));n" + ")n";
Well, that's a completely different matter! With minor changes (adding pinch zooming and finger rotation), I showed this program to my friends and colleagues, and at the same time asked how many triangles they thought there were in this scene. The results varied, and the question itself raised suspicions of a trick (in this case, respondents joked “one,” which was not far from the truth), but the correct answer consistently surprised. And everyone, as one, asked: why can a sphere be rotated around one axis, but not tilted?.. Hmm.
Incline
But the fact is that the slope in this scheme is much more difficult to implement. In fact, the task is not insurmountable, and I even coped with it, but there were some nuances.
In essence, the task boils down to taking the shifted coordinate V, while the coordinate U does not change: this happens because we add rotation around the axis X. The plan is this: we convert texture coordinates into screen coordinates (in the range [-1..1]), apply to them a rotation matrix around the horizontal axis (for this we write in advance in a new constant uTilt sine and cosine of the angle of inclination), and then we will use the new coordinate Y for sampling in our template texture. "Rotated" coordinate Z It will also be useful for us, with its help we will mirror the longitude for the back side of the ball). Screen coordinate Z you will have to calculate it explicitly so as not to make two texture samples from one texture, at the same time this will increase its accuracy.
Private final String quadFS = "precision mediump float;n" + "uniform sampler2D uTexture0;n" + "uniform sampler2D uTexture1;n" + "uniform float uOffset;n" + "uniform vec2 uTilt;n" + "varying vec4 TexCoord0; n" + "void main() (n" + " float sx = 2.0 * TexCoord0.x - 1.0;n" + " float sy = 2.0 * TexCoord0.y - 1.0;n" + " float z2 = 1.0 - sx * sx - sy * sy;n" + " if (z2 > 0.0) (;n" + " float sz = sqrt(z2);n" + " float y = (sy * uTilt.y - sz * uTilt.x + 1.0) * 0.5;n" + " float z = (sy * uTilt.x + sz * uTilt.y);n" + " vec4 vTex = texture2D(uTexture0, vec2(TexCoord0.x, y));n" + " vec3 vOff = vTex.xyz * 255.0;n" + " float hiY = floor(vOff.y / 16.0);n" + " float loY = vOff.y - 16.0 * hiY;n" + " vec2 vCoord = vec2( n" + " (256.0 * loY + vOff.x) / 4095.0,n" + " (vOff.z * 16.0 + hiY) / 4095.0);n" + " if (z< 0.0) { vCoord.x = 1.0 - vCoord.x; }n" + " vCoord.x += uOffset;n" + " vec3 vCol = texture2D(uTexture1, vCoord).rgb;n" + " gl_FragColor = vec4(vCol * sz, 1.0);n" + " } else {n" + " gl_FragColor = vec4(0.0, 0.0, 0.0, 0.0);n" + " }n" + "}n";
Hurray, the tilt was a success! But the strange noise at the border of the hemispheres is a little confusing. Alas, I have not yet been able to cope with this. Obviously, the problem lies in the insufficient addressing accuracy at the boundary points (the points on the circle itself correspond to too large a range of coordinates, one texel spreads over an interval of quite a noticeable length), and it’s unlikely that anything can be done about it. Well, but you can zoom in and scroll the ball almost the same way as in Google Earth. With the difference that here there are only two triangles.
For a long time I had thoughts about building my own house, but somehow in the form of interesting ideas that I noticed from others in life or in the media. Here I imagined what a house would look like that embodied all these ideas - a fox hole (dugout) turning into a mirror sphere hanging on a tree: D. In general, an ideological transformer, both outside and inside.
Now I am interested in geodesic domes and technologies for applying these principles for the construction of residential buildings and other useful and industrial structures (for example, sheds, bathhouses, greenhouses, sheds, workshops, hangars).
This summer (2011) I had the opportunity to observe it live, and even helped a little in the construction of a residential geodesic dome (photo on the left).
And now I came across interesting information on them, dug in, and decided to write an article for the future... a kind of cheat sheet so that I could quickly remember and find it. So, as information becomes available, I will update the article. I am sure it will be useful for the readers of the site.
Here they are:
Briefly about the history and what “geodesic” means.
As usual, everything new is well forgotten old.
Geo- our globe Earth
Remaining on D... - divide (the ancient Greeks divided and measured it... and not only them)
So, if you don’t go into the spatial and differential geometry of curved spaces))), then this is a dome made from part of a sphere, or rather a spherical polyhedron, since the earth is measured by points on its surface, which in our case are the vertices of this polyhedron. An important feature is the optimally distributed arrangement of vertices and faces tending to an ideal sphere. It is usually built on the basis of an icosahedron (20 triangular faces) or a dodecahedron (12 pentagonal faces).
Continued on next page.
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[comments/discussion]
| Vladimir (20:06 05.10.2016) Andrey, thank you for the interesting idea and useful tips! I am a geologist and geophysicist by education, sometimes I draw pictures and cut wood. This kind of workshop house, illuminated from all sides, is perhaps the best fit! And it reminds me of a crystal. In the evenings, looking at the stars, you can dream about flying on your own “UFO” in some next life. :-)) |
| Vyacheslav (18:10 11/14/2015) Looking for a job! 10 years of experience in low-rise construction. Design and construction of unusually shaped structures (geodesic domes). Three houses were built according to independent projects, one of which he built himself. Design of utilities (electrical, water supply, sewerage, roofing, insulation). I am very interested in alternative energy sources and the autonomy of residential buildings. We train quickly. Communicable. Punctual. Mobile. Portfolio available! |
| radius (02:20 11/28/2014) for those interested - the most comprehensive Russian-language resource on domes forum.domesworld.ru |
| Andrew (19:46 03.12.2013) to Mikhail Good afternoon. I see three reasons: + mainly people who are interested in geodesics prefer to use natural materials and products whenever possible; when building an Eco-house, polyurethane foam (PUF) is nonsense (PUF is considered a harmful polymer and you need to know how to build safely using it); + some technological difficulties and higher financial costs; + for such a steam room you need to use proper ventilation, in the opinion of the majority - forced |
| Mikhail (12:47 03.12.2013) Good afternoon I am amazed by the fact that after looking through many photo reports on the construction of domed houses, I never discovered the use of PPU spraying. On the contrary, everyone is suffering, stuffing this non-triangular mine. cotton wool into triangles, etc., they suffer from the vapor barrier bristling on the rounded surface. I can't understand why this is so. In simple frame construction, polyurethane foam is used everywhere, but here there is such neglect. Although some connectors are foamed with cylinders and windows and doors are placed on foam))) It seems to me that polyurethane foam and domed housing construction should be “all water” Or are there some peculiarities and the impossibility of using foamed polyurethane foam? |
| Andrew (08:38 09/24/2013) The triangles are assembled from boards using screws, and the triangles are assembled together using bolts. |
| Amir (10:09 09/23/2013) ... that geodesic dome, in the construction of which you helped, in the very first photo in the article - explain, or maybe you can send me information on the methods of joining (attaching) the frame elements of the dome to my email address. I will be very grateful. |
| adam gagarin (13:14 10/30/2012) We have not renewed Gravitonium ru for a long time, but all information about domes is available at www.valpak.ru & www.cupulageodesica.com/ru We ended up using thin steel thermo-reflective foil, glued directly to the inner surfaces of the plywood triangles. Thermos effect, weight like on the ISS, and heat and cold are reflected 99%. We will be happy to share information on all questions Sincerely, |
| Andrew (20:31 02/18/2012) Here's what they say in the Timberline FAQs: "The most common choices are fiberglass or rigid foam. Timberline"s 2" x 6" framing members allow for 5 1/2" of insulation, sufficient for most climatic conditions. Other options include spray-in expanding foam which is very effective. " "By using an expanding spray in foam insulation, it seals up the dome so well that no interior vapor barrier is needed." http://www.domehome.com/faqs.html So I haven’t studied the issue specifically about foam. Yes, this is that geodome. |
| staging (08:53 02/18/2012) Thanks for the answer. I really didn't want foam. Better than mineral wool. But if foam, then what kind? And here’s a photo http://www.zidar.ru/2011/09/stroim-kryishu-chast-vtoraya/#more-203 and the fact that where you have straight slats in the ventilation gap are they from one object? |
| Andrew (03:56 02/08/2012) - Timberline Geodesics usually uses glass wool and “construction” foam; - when using 2*6 inch ribs (approximately 50*150 mm), no ventilation gap is made and all voids are filled with foam, and they say that condensation does not form and vapor barrier is not required; - when using ribs of larger sections (50*200/300), as an option, they offer to make cuts similar to Natural Spaces Domes; - the roof consists of triangular edges covered with an under-roofing carpet and topped with bitumen/wood/metal tiles or a special coating is used. So you can try to blow everything out with foam or do it according to the classic maximum scheme with a ventilation gap: |
| staging (00:48 02/08/2012) I am interested in a ready-made solution for ventilation of the under-roof space and insulation. Roofing pie. Frame 3v 5/8 TIMBERLINE. |
Spherical triangle and its applications.
Spherical triangle- a geometric figure on the surface of a sphere formed by the intersection of three large circles. Three large circles on the surface of a sphere that do not intersect at one point form eight spherical triangles. A spherical triangle whose sides are all less than half a great circle is called Eulerian.
The side of a spherical triangle is measured by the size of the central angle resting on it. The angle of a spherical triangle is measured by the size of the dihedral angle between the planes in which the sides of this angle lie. Spherical trigonometry studies the relationships between the elements of spherical triangles.
Properties of a spherical triangle:
- In addition to the three criteria for the equality of plane triangles, one more is true for spherical triangles: two spherical triangles are equal if their corresponding angles are equal.
- For the sides of a spherical triangle, the 3 triangle inequalities hold: each side is less than the sum of the other two sides and greater than their difference.
- The sum of all sides a + b + c is always less than 2πR.
- The quantity 2πR − (a + b + c) is called the spherical defect
- The sum of the angles of a spherical triangle s = α + β + γ is always less than 3π and greater than π
- The quantity is called spherical excess or spherical kurtosis
- The area of a spherical triangle is determined by the formula.
- Unlike a flat triangle, a spherical triangle can have two or even three angles of 90° each.
Among all spherical polygons, the spherical triangle is of greatest interest. Three large circles, intersecting in pairs at two points, form eight spherical triangles on the sphere. Knowing the elements (sides and angles) of one of them, it is possible to determine the elements of all the others, so we consider the relationships between the elements of one of them, the one whose all sides are less than half of the great circle. The sides of a triangle are measured by the plane angles of the trihedral angle OABC, the angles of the triangle are measured by the dihedral angles of the same trihedral angle, cm in Fig.
The properties of spherical triangles differ in many ways from the properties of triangles on a plane. Thus, to the known three cases of equality of rectilinear triangles, a fourth is added: two triangles ABC and A'B'C' are equal if the three angles RA = RA', PB = PB', RS = RS' are equal, respectively. Thus, there are no similar triangles on the sphere; moreover, in spherical geometry there is no very concept of similarity, because There are no transformations that change all distances by the same (not equal to 1) number of times. These features are associated with a violation of the Euclidean axiom of parallel lines and are also inherent in Lobachevsky’s geometry. Triangles that have equal elements and different orientations are called symmetrical, such as, for example, triangles AC'C and BCC' The sum of the angles of any spherical triangle is always greater than 180°. The difference RA + PB + RS – p = d (measured in radians) is a positive quantity and is called the spherical excess of a given spherical triangle. Area of a spherical triangle: S = R2 d where R is the radius of the sphere and d is the spherical excess. This formula was first published by the Dutchman A. Girard in 1629 and named after him.
If we consider a digon with angle a, then at 226 = 2p/n (n is an integer), the sphere can be cut into exactly n copies of such a diagon, and the area of the sphere is 4nR2 = 4p at R = 1, so the area of the diagon is 4p/n = 2a. This formula is also true for a = 2pt/n and, therefore, is true for all a. If we continue the sides of the spherical triangle ABC and express the area of the sphere in terms of the areas of the resulting bigons with angles A, B, C and its own area, then we can arrive at the above Girard formula.
A spherical triangle means a triangle on the surface of a sphere, composed of arcs of great circles - that is, circles whose center is the center of the sphere. The angles of a spherical triangle are the angles between the tangents to its sides drawn at its vertices. Like the angles of a regular triangle, they vary from 0 to 180°. Unlike a flat triangle, a spherical triangle has a sum of angles that is not equal to 180°, but greater: this is easy to verify by considering, for example, a triangle formed by the arcs of two meridians and the equator on the globe: although the meridians converge at the pole, both of them are perpendicular to the equator, and This means that this triangle has two right angles! A spherical triangle can have two right angles
Already among the Indian Varahamihira (V-VI centuries), among Arab mathematicians and astronomers starting from the 9th century. (Sabit ibn Korra, al-Battani), and among Western mathematicians, starting from Regiomontanus (XV century), a remarkable theorem about spherical triangles is found in various formulations. Here's how it can be formulated in modern notation:
cosa = cosbcosc + sinbsinccosA. The spherical cosine theorem is very important for both astronomy and geography. This theorem allows you to use the coordinates of two cities A and B to find the distance between them. In addition, the spherical theorem of cosines helped mathematicians in Islamic countries in solving another practical problem: in a city with given coordinates, find the direction to the holy city of Mecca (every devout Muslim must pray in the direction of Mecca five times a day). When solving this problem, considering city B to be Mecca, it was necessary to find angle A of the same triangle.
Page from the “Collected Rules of the Science of Astronomy,” 11th century, author unknown. In astronomy, the spherical cosine theorem allows one to move from one coordinate system on the celestial sphere to another. Most often, three such systems are used: in one, the celestial equator serves as the equator, and the poles are the poles of the world, around which the visible daily rotation of the luminaries occurs; in another, the equator is the ecliptic - the circle along which the visible movement of the Sun takes place during the year against the background of stars; in the third, the role of the equator is played by the horizon, and the role of the poles is played by zenith and nadir. In particular, thanks to the spherical cosine theorem, it is possible to calculate the height of the Sun above the horizon at different times and on different days of the year.
Sails in architecture are a spherical triangle, providing a transition from the square under-dome space to the circumference of the dome. Sail, pandative (from the French pendentif) - part of the vault, an element of the dome structure, through which the transition is made from the rectangular base to the dome floor or its drum. The sail has the shape of a spherical triangle, with its apex downward, and fills the space between the girth arches connecting the adjacent pillars of the domed square. The bases of the spherical triangles of the sails together form a circle and distribute the load of the dome along the perimeter of the arches.
Dome on sails 
Sail painting 
George Nelson
"The designer can relax a little and have fun; the result can be a joke, amusement. It is surprising how often this can be very significant amusement" George Nelson
George Nelson is an American designer, architect, critic and design theorist. (1908, Hartford, Connecticut - 1986, New York)
He designed lighting fixtures, clocks, furniture, packaging, and was involved in exhibition design.
George Nelson's most famous design projects represent a masterly stylization of geometric shapes in the spirit of op art or geometric abstractionism.
The designer bases the shape of his famous black chair on the basis of a spherical triangle, which was widely used in the architectural designs of domed structures. In particular, in Byzantine and Russian churches such a spherical triangle was called a “sail”. Thanks to the “sail” there was a smooth transition from the under-dome support to the dome.
George Nelson (1908-1986) Escher engraving
Concentric spheres. 1935. End engraving 24 by 24 cm. Four hollow concentric spheres are illuminated by a central light source. Each sphere is composed of a grid formed by nine large intersecting rings; they divide the spherical surface into 48 similar spherical triangles. Maurits Cornelis Escher (Dutch: Maurits Cornelis June 17, 1898, Leeuwarden, Netherlands - March 27, 1972, Laren, Netherlands) - Dutch graphic artist.
Application of spherical triangle:
- Using spherical triangles in 3D graphics
- In astronomy
- In geography. The spherical triangle theorem allows you to use the coordinates of two cities A and B to find the distance between them.
- In architecture
- Chair design by George Nelson
- In engraving
Spherical triangles.
On the surface of a ball, the shortest distance between two points is measured along the circumference of a great circle, that is, a circle whose plane passes through the center of the ball. The vertices of a spherical triangle are the intersection points of three rays emanating from the center of the ball and the spherical surface. Parties a, b, c A spherical triangle is defined as those angles between the rays that are less than 180°. Each side of the triangle corresponds to an arc of a great circle on the surface of the ball (Fig. 1). Angles A, B, C spherical triangle, opposite sides a, b, c accordingly, they are, by definition, angles less than 180° between arcs of great circles corresponding to the sides of a triangle, or angles between planes defined by these rays.
Properties of spherical triangles.
Each side and angle of a spherical triangle is, by definition, less than 180°. The geometry on the surface of the ball is non-Euclidean; In every spherical triangle, the sum of the sides is between 0 and 360°, the sum of the angles is between 180° and 540°. In each spherical triangle, the larger angle lies opposite the larger side. The sum of any two sides is greater than the third side, and the sum of any two angles is less than 180° plus the third angle.
A spherical triangle is uniquely defined (up to symmetry transformation):
- three sides,
- three corners,
- two sides and the angle between them,
- side and two adjacent corners.
Solving spherical triangles (Table)
(see formulas below and Fig. 1 above)
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Calculation formulas |
Conditions for the existence of a solution |
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| 1 |
Three sides a, b, c |
A, B, C |
The sum of two sides must be greater than the third |
| 2 |
A, B, C |
a, b, c from (8) and cyclic permutation |
The sum of two angles must be less than 180° plus the third angle |
| 3 |
Two sides and the angle between them b, c, A |
from (6), then IN And WITH; A from (7), (8) or (4) |
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| 4 |
Two angles and the side between them B, C, a |
from (6), then b And With; A from (7), (8) or (5) |
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| 5 |
Two sides and the angle opposite one of them b, s, b |
WITH from (3); A And A from (6) |
sin With sin IN≤ sin b. Those of the quantities are saved With, for which A - B And a - b have the same sign; A+B- 180° And a + b- 180° |
| 6 |
Two angles and the side opposite one of them B, C, b |
With from (3); A And A from (6) |
A problem has one or two solutions if sin b sin WITH≤ sin IN. Those of the quantities are saved With, for which A - B And a - b have the same sign; A+B- 180° And a + b- 180° must also be of the same sign |
Formulas for solving spherical triangles
In the following ratios A, B, C are angles opposite to the sides respectively a, b, c spherical triangle. The “radii” of the circumscribed and inscribed cones are designated by r and r, respectively. Formulas not included in the list can be obtained by simultaneous cyclic permutation A, B, C And a, b, c. The table above allows you to calculate the sides and angles of any spherical triangle given the three appropriately given sides and/or angles. The inequalities noted at the beginning of paragraph 2 must be taken into account in order to exclude extraneous results when solving triangles.
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sine theorem |
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cosine theorem for sides |
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cosine theorem for angles |
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Napier's analogies |
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Analogies of Delambre and Gauss |
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Thus, if tables of the hav function are available, then these formulas can be used to solve spherical triangles: |
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Other similar relations can be obtained by cyclic permutation |
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