The origins of spherical trigonometry inGreek mathematics and the major developments in Islamic mathematics are discussed fully inHistory of trigonometry andMathematics in medieval Islam. The subject came to fruition in Early Modern times with important developments byJohn Napier,Delambre and others, and attained an essentially complete form by the end of the nineteenth century with the publication ofIsaac Todhunter's textbookSpherical trigonometry for the use of colleges and Schools.[1]Since then, significant developments have been the application of vector methods,quaternion methods, and the use of numerical methods.
Such polygons may have any number of sides greater than 1. Two-sided spherical polygons—lunes, also calleddigons orbi-angles—are bounded by two great-circle arcs: a familiar example is the curved outward-facing surface of a segment of an orange. Three arcs serve to define a spherical triangle, the principal subject of this article. Polygons with higher numbers of sides (4-sided spherical quadrilaterals, 5-sided spherical pentagons, etc.) are defined in similar manner. Analogously to their plane counterparts, spherical polygons with more than 3 sides can always be treated as the composition of spherical triangles.
One spherical polygon with interesting properties is thepentagramma mirificum, a 5-sided sphericalstar polygon with a right angle at every vertex.
From this point in the article, discussion will be restricted to spherical triangles, referred to simply astriangles.
Both vertices and angles at the vertices of a triangle are denoted by the same upper case lettersA,B, andC.
Sides are denoted by lower-case letters:a,b, andc. The sphere has a radius of 1, and so the side lengths and lower case angles are equivalent (seearc length).
TheangleA (respectively,B andC) may be regarded either as thedihedral angle between the two planes that intersect the sphere at thevertexA, or, equivalently, as the angle between thetangents of the great circle arcs where they meet at the vertex.
Angles are expressed inradians. The angles ofproper spherical triangles are (by convention) less thanπ, so that(Todhunter,[1] Art.22,32).
In particular, the sum of the angles of a spherical triangle is strictly greater than the sum of the angles of a triangle defined on the Euclidean plane, which is always exactlyπ radians.
Sides are also expressed in radians. A side (regarded as a great circle arc) is measured by the angle that it subtends at the centre. On theunit sphere, this radian measure is numerically equal to the arc length. By convention, the sides ofproper spherical triangles are less thanπ, so that(Todhunter,[1] Art.22,32).
The sphere's radius is taken as unity. For specific practical problems on a sphere of radiusR the measured lengths of the sides must be divided byR before using the identities given below. Likewise, after a calculation on the unit sphere the sidesa,b, andc must be multiplied by R.
Thepolar triangle associated with a triangle△ABC is defined as follows. Consider the great circle that contains the side BC. This great circle is defined by the intersection of adiametral plane with the surface. Draw the normal to that plane at the centre: it intersects the surface at two points and the point that is on the same side of the plane asA is (conventionally) termed the pole ofA and it is denoted byA'. The pointsB' andC' are defined similarly.
The triangle△A'B'C' is the polar triangle corresponding to triangle △ABC. The angles and sides of the polar triangle aregiven by (Todhunter,[1] Art.27)Therefore, if any identity is proved for△ABC then we can immediately derive a second identity by applying the first identity to the polar triangle by making the above substitutions. This is how the supplemental cosine equations are derived from the cosine equations. Similarly, the identities for a quadrantal triangle can be derived from those for a right-angled triangle. The polar triangle of a polar triangle is the original triangle.
If the3 × 3 matrixM has the positionsA,B, andC as its columns then the rows of the matrix inverseM−1, if normalized to unit length, are the positionsA′,B′, andC′. In particular, when△A′B′C′ is the polar triangle of△ABC then△ABC is the polar triangle of△A′B′C′.
The cosine rule is the fundamental identity of spherical trigonometry: all other identities, including the sine rule, may be derived from the cosine rule:
These identities generalize the cosine rule of planetrigonometry, to which they are asymptotically equivalentin the limit of small interior angles. (On the unit sphere, if set and etc.; seeSpherical law of cosines.)
The sphericallaw of sines is given by the formulaThese identities approximate the sine rule of planetrigonometry when the sides are much smaller than the radius of the sphere.
The spherical cosine formulae were originally proved by elementary geometry and the planar cosine rule (Todhunter,[1] Art.37). He also gives a derivation using simple coordinate geometry and the planar cosine rule (Art.60). The approach outlined here uses simplervector methods. (These methods are also discussed atSpherical law of cosines.)
Consider three unit vectorsOA→,OB→,OC→ drawn from the origin to the vertices of the triangle (on the unit sphere). The arcBC subtends an angle of magnitudea at the centre and thereforeOB→ ·OC→ = cosa. Introduce a Cartesian basis withOA→ along thez-axis andOB→ in thexz-plane making an anglec with thez-axis. The vectorOC→ projects toON in thexy-plane and the angle betweenON and thex-axis isA. Therefore, the three vectors have components:
The scalar productOB→ · OC→ in terms of the components isEquating the two expressions for the scalar product givesThis equation can be re-arranged to give explicit expressions for the angle in terms of the sides:
The other cosine rules are obtained by cyclic permutations.
This derivation is given in Todhunter,[1] (Art.40). From the identity and the explicit expression forcosA given immediately aboveSince the right hand side is invariant under acyclic permutation ofa,b, andc the spherical sine rule follows immediately.
There are many ways of deriving the fundamental cosine and sine rules and the other rules developed in the following sections. For example, Todhunter[1] gives two proofs of the cosine rule (Articles 37 and 60) and two proofs of the sine rule (Articles 40 and 42). The page onSpherical law of cosines gives four different proofs of the cosine rule. Text books on geodesy[2] and spherical astronomy[3] give different proofs and the online resources ofMathWorld provide yet more.[4] There are even more exotic derivations, such as that of Banerjee[5] who derives the formulae using thelinear algebra of projection matrices and also quotes methods indifferential geometry and thegroup theory of rotations.
The derivation of the cosine rule presented above has the merits of simplicity and directness and the derivation of the sine rule emphasises the fact that no separate proof is required other than the cosine rule. However, the above geometry may be used to give an independent proof of the sine rule. Thescalar triple product,OA→ · (OB→ ×OC→) evaluates tosinb sinc sinA in the basis shown. Similarly, in a basis oriented with thez-axis alongOB→, the triple productOB→ · (OC→ ×OA→), evaluates tosinc sina sinB. Therefore, the invariance of the triple product under cyclic permutations givessinb sinA = sina sinB which is the first of the sine rules. See curved variations of thelaw of sines to see details of this derivation.
When any three of the differentialsda,db,dc,dA,dB,dC are known, the following equations, which are found by differentiating the cosine rule and using the sine rule, can be used to calculate the other three by elimination:[6]
The six parts of a triangle may be written incyclic order as (aCbAcB). The cotangent, or four-part, formulae relate two sides and two angles forming fourconsecutive parts around the triangle, for example (aCbA) orBaCb). In such a set there are inner and outer parts: for example in the set (BaCb) the inner angle isC, the inner side isa, the outer angle isB, the outer side isb. The cotangent rule may be written as (Todhunter,[1] Art.44)and the six possible equations are (with the relevant set shown at right):To prove the first formula start from the first cosine rule and on the right-hand side substitute forcosc from the third cosine rule:The result follows on dividing bysina sinb. Similar techniqueswith the other two cosine rules give CT3 and CT5. The other three equations follow by applying rules 1, 3 and 5 to the polar triangle.
Another twelve identities follow by cyclic permutation.
Todhunter[7] (Art 45) derives the half angle formulas for the angles and sides in terms of the sides and angles, respectively. His book is available as an ebook in the public domain from Project Gutenberg. The first equation may be proved by using the law of cosines for side a in terms of sides b and c and angle A, by using the identity and by expressing the product of two sines as half the difference of the cosine of their angle difference angle minus the cosine of their angle sum (Seesum-to-product identities). In detail:
The second formula uses the identity the third is a quotient and the remainder follow by applying the results to the polar triangle.
Another eight identities follow by cyclic permutation.
These identities follow by division of the Delambre formulae. (Todhunter,[1] Art.52)
In particular, the first and second Napier formulas are useful in solving a spherical triangle when three of a b A B are given but not c or C. These two formulas give in terms of a b A B. This spherical triangle cannot be solved straightforwardly using only the cosine and sine laws.
When one of the angles, sayC, of a spherical triangle is equal toπ/2 the various identities given above are considerably simplified. There are ten identities relating three elements chosen from the seta,b,c,A, andB.
Napier[10] provided an elegantmnemonic aid for the ten independent equations: the mnemonic is called Napier's circle or Napier's pentagon (when the circle in the above figure, right, is replaced by a pentagon).
First, write the six parts of the triangle (three vertex angles, three arc angles for the sides) in the order they occur around any circuit of the triangle: for the triangle shown above left, going clockwise starting witha givesaCbAcB. Next replace the parts that are not adjacent toC (that isA,c, andB) by their complements and then delete the angleC from the list. The remaining parts can then be drawn as five ordered, equal slices of a pentagram, or circle, as shown in the above figure (right). For any choice of three contiguous parts, one (themiddle part) will be adjacent to two parts and opposite the other two parts. The ten Napier's Rules are given by
sine of the middle part = the product of the tangents of the adjacent parts
sine of the middle part = the product of the cosines of the opposite parts
The key for remembering which trigonometric function goes with which part is to look at the first vowel of the kind of part: middle parts take the sine, adjacent parts take the tangent, and opposite parts take the cosine.For an example, starting with the sector containinga we have:The full set of rules for the right spherical triangle is (Todhunter,[1] Art.62)
A quadrantal spherical triangle together with Napier's circle for use in his mnemonics
A quadrantal spherical triangle is defined to be a spherical triangle in which one of the sides subtends an angle ofπ/2 radians at the centre of the sphere: on the unit sphere the side has lengthπ/2. In the case that the sidec has lengthπ/2 on the unit sphere the equations governing the remaining sides and angles may be obtained by applying the rules for the right spherical triangle of the previous section to the polar triangle△A'B'C' with sidesa', b', c' such thatA'=π −a,a'=π −A etc. The results are:
Multiplying the first cosine rule bycosA givesSimilarly multiplying the first supplementary cosine rule bycosa yieldsSubtracting the two and noting that it follows from the sine rules that produces Cagnoli's equationwhich is a relation between the six parts of the spherical triangle.[11]
The solution of triangles is the principal purpose of spherical trigonometry: given three, four or five elements of the triangle, determine the others. The case of five given elements is trivial, requiring only a single application of the sine rule. For four given elements there is one non-trivial case, which is discussed below. For three given elements there are six cases: three sides, two sides and an included or opposite angle, two angles and an included or opposite side, or three angles. (The last case has no analogue in planar trigonometry.) No single method solves all cases. The figure below shows the seven non-trivial cases: in each case the given sides are marked with a cross-bar and the given angles with an arc. (The given elements are also listed below the triangle). In the summary notation here such as ASA, A refers to a given angle and S refers to a given side, and the sequence of A's and S's in the notation refers to the corresponding sequence in the triangle.
Case 1: three sides given (SSS). The cosine rule may be used to give the anglesA,B, andC but, to avoid ambiguities, the half angle formulae are preferred.
Case 2: two sides and an included angle given (SAS). The cosine rule givesa and then we are back to Case 1.
Case 3: two sides and an opposite angle given (SSA). The sine rule givesC and then we have Case 7. There are either one or two solutions.
Case 4: two angles and an included side given (ASA). The four-part cotangent formulae for sets (cBaC) and (BaCb) givec andb, thenA follows from the sine rule.
Case 5: two angles and an opposite side given (AAS). The sine rule givesb and then we have Case 7 (rotated). There are either one or two solutions.
Case 6: three angles given (AAA). The supplemental cosine rule may be used to give the sidesa,b, andc but, to avoid ambiguities, the half-side formulae are preferred.
Case 7: two angles and two opposite sides given (SSAA). Use Napier's analogies fora andA; or, use Case 3 (SSA) or case 5 (AAS).
The solution methods listed here are not the only possible choices: many others are possible. In general it is better to choose methods that avoid taking an inverse sine because of the possible ambiguity between an angle and its supplement. The use of half-angle formulae is often advisable because half-angles will be less thanπ/2 and therefore free from ambiguity. There is a full discussion in Todhunter. The articleSolution of triangles#Solving spherical triangles presents variants on these methods with a slightly different notation.
There is a full discussion of the solution of oblique triangles in Todhunter.[1]: Chap. VI See also the discussion in Ross.[12]Nasir al-Din al-Tusi was the first to list the six distinct cases (2–7 in the diagram) of aright triangle in spherical trigonometry.[13]
Another approach is to split the triangle into two right-angled triangles. For example, take the Case 3 example whereb,c, andB are given. Construct the great circle fromA that is normal to the sideBC at the pointD. Use Napier's rules to solve the triangle△ABD: usec andB to find the sidesAD andBD and the angle∠BAD. Then use Napier's rules to solve the triangle△ACD: that is useAD andb to find the sideDC and the anglesC and∠DAC. The angleA and sidea follow by addition.
Not all of the rules obtained are numerically robust in extreme examples, for example when an angle approaches zero or π. Problems and solutions may have to be examined carefully, particularly when writing code to solve an arbitrary triangle.
Lexell's theorem: the triangles of constant area on a fixed baseAB have their free vertexC along asmall circle through the points antipodal toA andB.
Consider anN-sided spherical polygon and letAn denote then-thinterior angle. The area of such a polygon is given by (Todhunter,[1] Art.99)
Via triangulation of the spherical polygon, the proof of this theorem can be reduced to a proof for a spherical triangle.For the case of a spherical triangle with anglesA,B, andC this isGirard's theoremwhereE is the amount by which the sum of the angles exceedsπ radians, called thespherical excess of the triangle. This theorem is named after its author,Albert Girard.[14] An earlier proof was derived, but not published, by the English mathematicianThomas Harriot in 1603.[15] On a sphere of radiusR both of the above area expressions are multiplied byR2. The definition of the excess is independent of the radius of the sphere.
The converse result may be written as
Since the area of a triangle cannot be negative the spherical excess is always positive. It is not necessarily small, because the sum of the angles may attain 5π (3π forproper angles). For example,an octant of a sphere is a spherical triangle with three right angles, so that the excess isπ/2. In practical applications itis often small: for example the triangles of geodetic survey typically have a spherical excess much less than 1' of arc.[16] On the Earth the excess of anequilateral triangle with sides 21.3 km (and area 393 km2) is approximately 1 arc second.
There are many formulae for the excess. For example, Todhunter,[1] (Art.101—103) gives ten examples including that ofL'Huilier:where. This formula is reminiscent ofHeron's formula for planar triangles.
Because some triangles are badly characterized bytheir edges (e.g., if), it is often better to usethe formula for the excess in terms of two edges and their included angle
When triangle△ABC is a right triangle with right angle atC, thencosC = 0 andsinC = 1, so this reduces to
The spherical excess of a spherical quadrangle bounded by the equator, the two meridians of longitudes and and the great-circle arc between two points with longitude and latitude and is
This result is obtained from one of Napier's analogies. In the limit where are all small, this reduces to the familiar trapezoidal area,.
The area of a polygon can be calculated from individual quadrangles of the above type, from (analogously) individual triangle bounded by a segment of the polygon and two meridians,[17] by aline integral withGreen's theorem,[18] or via anequal-area projection as commonly done in GIS. The other algorithms can still be used with the side lengths calculated using agreat-circle distance formula.
^Todhunter, Isaac (1873). "Note on the history of certain formulæ in spherical trigonometry".The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science.45 (298):98–100.doi:10.1080/14786447308640820.
^Ross, Debra Anne.Master Math: Trigonometry, Career Press, 2002.
^O'Connor, John J.;Robertson, Edmund F.,"Nasir al-Din al-Tusi",MacTutor History of Mathematics Archive,University of St Andrews "One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth"
^Chamberlain, Robert G.; Duquette, William H. (17 April 2007).Some algorithms for polygons on a sphere. Association of American Geographers Annual Meeting. NASA JPL. Retrieved7 August 2020.
Some Algorithms for Polygons on a Sphere Robert G. Chamberlain, William H. Duquette, Jet Propulsion Laboratory. The paper develops and explains many useful formulae, perhaps with a focus on navigation and cartography.
