Computer Code

The spherical and plane trigonometric laws pertaining to a triangle: cosine, haversine, sine (long and short forms), tangent, half-tangent, Napier (one and two), and area are provided as functions, which may be cut and pasted into an application.

The following code is intended for Maple(r) 7

Instructions

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You have all of the functions life, in Maple!

> # COMMON functions

The sides of the triangle are {sa, sb, sc} and the opposite angles are {aA, aB, aC}.

for brevity, a few functions

> ss:=unapply((sa+sb+sc)/2,(sa,sb,sc));

> aS:=unapply((aA+aB+aC)/2,(aA,aB,aC));

> sr:=unapply(sqrt(sin(ss(sa,sb,sc)-sa)*sin(ss(sa,sb,sc)-sb)*sin(ss(sa,sb,sc)-sc)/sin(ss(sa,sb,sc))),(sa,sb,sc));

> aR:=unapply(sqrt(cos(aS(aA,aB,aC)-aA)*cos(aS(aA,aB,aC)-aB)*cos(aS(aA,aB,aC)-aC)/cos(Pi-aS(aA,aB,aC))),(aA,aB,aC));

definition of the haversine function and its inverse

> hav:=unapply((sin(x/2))^2,x);

> archav:=unapply(2*arcsin(sqrt(x)),x);

the archav above is the principal-value. the other value is its negative.

[place holder]

> # SPHERICAL laws

This is the spherical section.

spherical law of cosines. Caution: do NOT use these equations for numerical calculation! Instead, use the law of haversines or its dual.

> law_cos_sa:=unapply(arccos(sin(sb)*sin(sc)*cos(aA)+cos(sb)*cos(sc)),(sb,sc,aA));

> law_cos_aA:=unapply(arccos((cos(sa)-cos(sb)*cos(sc))/(sin(sb)*sin(sc))),(sa,sb,sc));

and its dual

> law_cosdual_aA:=unapply(arccos(sin(aB)*sin(aC)*cos(sa)-cos(aB)*cos(aC)),(aB,aC,sa));

> law_cosdual_sa:=unapplly(arccos((cos(aA)+cos(aB)*cos(aC))/(sin(aB)*sin(aC))),(aA,aB,aC));

spherical law of haversines

> law_hav_sa:=unapply(archav(hav(sb-sc)+sin(sb)*sin(sc)*hav(aA)),(sb,sc,aA));

> law_hav_aA:=unapply(archav((hav(sa)-hav(sb-sc))/(sin(sb)*sin(sc))),(sa,sb,sc));

and its dual

> law_havdual_aA:=unapply(Pi-archav(hav(aB-aC)+sin(aB)*sin(aC)*hav(Pi-sa)),(aB,aC,sa));

> law_havdual_sa:=unapply(Pi-archav((hav(Pi-aA)-hav(aB-aC))/(sin(aB)*sin(aC))),(aA,aB,aC));

each of the foregoing six functions is the principal-value. the other value is the negative of the principal-value.

spherical isosceles law of haversines

> law_isosceles_hav_sa:=unapply(2*arcsin(sin(sb)*sin(aA/2)),(sb,aA));

> las_isosceles_hav_aA:=unapply(2*arcsin(sin(sa/2)/sin(sb)),(sa,sb));

and its dual

> law_isosceles_havdual_aA:=unapply(2*arccos(sin(aB)*cos(sa/2)),(aB,sa));

> law_isosceles_havdual_sa:=unapply(2*arccos(cos(aA/2)/sin(aB)),(aA,aB));

spherical law of sines (short and long forms) and their duals. In the calling sequence of the short form only, the side sb and angle aB may be replaced by the side sc and angle aC, respectively

> law_sin_expr:=unapply(sqrt((2* cos(sa)* cos(sb)* cos(sc) + ((sin(sa))^2 + (sin(sb))^2 + (sin(sc))^2)^2 - 2)) / (sin(sa)* sin(sb) *sin(sc)),(sa,sb,sc));

> law_sin_short_aA:=unapply(arcsin(sin(sa)*sin(aB)/sin(sb)),(sa,aB,sb));

> law_sin_long_aA:=unapply(arcsin(sin(sa)*law_sin_expr(sa,sb,sc)),(sa,sb,sc));

> law_sin_short_sa:=unapply(arcsin(sin(aA)*sin(sb)/sin(aB)),(aA,aB,sb));

> law_sin_long_sa:=unapply(arcsin(sin(aA)*law_sin_expr(Pi-aA,Pi-aB,Pi-aC)),(aA,aB, aC));

each of the foregoing four functions is the principal-value. the other value is the supplement of the principal-value.

We may combine the haversine and sine (short form) laws, to obtain a consistent additional angle (or side, in the case of the dual).

> law_hav_and_sin_aB:=unapply(law_hav_aA(sb,sc,sa)*signum(law_sin_short_aA(sb,aA,sa)),(sa,sb,sc,aA));

> law_hav_and_sin_sb:=unapply(law_havdual_sa(aB,aC,aA)*signum(law_sin_short_sa(aB,aA,sa)),(aA,aB,aC,sa));

spherical law of tangents. Use as a check. The function should calculate to zero.

> law_tan:=unapply(tan((sb-sc)/2)/tan((sb+sc)/2)-tan((aB-aC)/2)/tan((aB+aC)/2),(sb,sc,aB,aC));

[place holder]

spherical law of half-tangent and its dual

> law_half_tan_aA:=unapply(2*arctan(sr(sa,sb,sc)/sin(ss(sa,sb,sc)-sa)),(sa,sb,sc));

> law_half_tan_sa:=unapply(2*arccot(aR(aA,aB,aC)/sin(aS(aA,aB,aC)-aA)),(aA,aB,aC));

[place holder]

spherical Napier's analogies and their duals

> napier_one_sc:=unapply(2*arctan(sin((aA+aB)/2)*tan((sa-sb)/2)/sin((aA-aB)/2)),(aA,aB,sa,sb));

> napier_one_aC:=unapply(2*arccot(sin((sa+sb)/2)*tan((aA-aB)/2)/sin((sa-sb)/2)),(sa,sb,aA,aB));

> napier_two_sc:=unapply(2*arctan(cos((aA+aB)/2)*tan((sa+sb)/2)/cos((aA-aB)/2)),(aA,aB,sa,sb));

> napier_two_aC:=unapply(2*arccot(cos((sa+sb)/2)*tan((aA+aB)/2)/cos((sa-sb)/2)),(sa,sb,aA,aB));

[place holder]

spherical area is equal to the spherical-excess E multiplied by the square of the radius of the sphere.

> spherical_excess_aE:=unapply(2*aS(aA,aB,aC)-Pi,(aA,aB,aC));

> spherical_lHuilier_sE:=unapply(4*arctan(sqrt(tan(ss(sa,sb,sc)/2)*tan((ss(sa,sb,sc)-sa)/2)*tan((ss(sa,sb,sc)-sb)/2)*tan((ss(sa,sb,sc)-sc)/2))),(sa,sb,sc));

[place holder]

> # PLANE laws

This is the plane section.

For brevity, the radii of the incricle and circumcircle.

> plane_sr:=unapply((ss(sa,sb,sc)-sa)*(ss(sa,sb,sc)-sb)*(ss(sa,sb,sc)-sc)/ss(sa,sb,sc),(sa,sb,sc));

> plane_aR:=unapply(sqrt((sa^2*sb^2*sc^2)/(4*sb^2*sc^2+sc^2*sa^2+sa^2*sb^2)-(sa^2+sb^2+sc^2)^2),(sa,sb,sc));

plane law of cosines. Caution: do NOT use these equations for numerical calculation! Instead, use the law of haversines or its dual.

> law_cos_plane_sa:=unapply(sqrt(sb^2+sc^2-2*sb*sc*cos(aA)),(sb,sc,aA));

> law_cos_plane_aA:=unapply(arccos((sb^2+sc^2-sa^2)/(2*sb*sc)),(sa,sb,sc));

and its dual

> law_cosdual_plane_aA:=unapply(aB+aC-Pi,(aB,aC));

plane law of haversines

> law_hav_plane_sa:=unapply(sqrt((sb-sc)^2+4*sb*sc*hav(aA)),(sb,sc,aA));

> law_hav_plane_aA:=unapply(archav((sa-(sb-sc))*(sa+(sb-sc))/(4*sb*sc)),(sa,sb,sc));

each of the foregoing four functions is the principal-value. the other value is the negative of the principal-value.

plane isosceles law of haversines

> law_isosceles_hav_plane_sa:=unapply(2*sb*sin(aA/2),(sb,aA));

> las_isosceles_hav_plane_aA:=unapply(2*arcsin(sa/(2*sb)),(sa,sb));

plane law of sines (short and long forms) and the dual of the short form, only. In the calling sequence of the short form only, the side sb and angle aB may be replaced by the side sc and angle aC, respectively.

> law_sin_short_plane_aA:=unapply(arcsin(sa*sin(aB)/sb),(sa,aB,sb));

> law_sin_long_plane_aA:=unapply(arcsin(sa/(2*plane_aR(sa,sb,sc))),(sa,sb,sc));

> law_sin_short_plane_sa:=unapply(arcsin(sin(aA)*sb/sin(aB)),(aA,aB,sb));

each of the foregoing four functions is the principal-value. the other value is the supplement of the principal-value.

plane law of tangents. Use as a check. The function should calculate to zero.

> law_plane_tan:=unapply((sb-sc)/(sb+sc)-tan((aB-aC)/2)/tan((aB+aC)/2),(sb,sc,aB,aC));

[place holder]

plane law of half-tangent and its dual

> law_half_tan_plane_aA:=unapply(2*arctan(plane_sr(sa,sb,sc)/(ss(sa,sb,sc)-sa)),(sa,sb,sc));

> law_half_tan_plane_sa:=unapply(2*sin(aS(aA,aB,aC)-aA)/plane_aR(aA,aB,aC),(aA,aB,aC));

each of the foregoing two functions is the principal-value. the other value is the negative of the principal-value.

[place holder]

plane Napier's analogies and their duals

> napier_one_plane_sc:=unapply(sin((aA+aB)/2)*(sa-sb)/sin((aA-aB)/2),(aA,aB,sa,sb));

> napier_one_plane_aC:=unapply(2*arccot((sa+sb)*tan((aA-aB)/2)/(sa-sb)),(sa,sb,aA,aB));

> napier_two_plane_sc:=unapply(cos((aA+aB)/2)*(sa+sb)/cos((aA-aB)/2),(aA,aB,sa,sb));

> napier_two_plane_aC:=unapply(Pi-(aA+aB),(aA,aB));

[place holder]

plane area may be computed by either of the following functions.

> plane_Heron_area:=unapply(sqrt(ss(sa,sb,sc)*(ss(sa,sb,sc)-sa)*(ss(sa,sb,sc)-sb)*(ss(sa,sb,sc)-sc)),(sa,sb,sc));

> plane_circum_area:=unapply((sa*sb*sc)/(4*aR(sa,sb,sc)),(sa,sb,sc));

> plane_in_area:=unapply(sr(sa,sb,sc)*ss(sa,sb,sc),(sa,sb,sc));

> plane_area:=unapply((sa*sb/2)*sin(aC),(sa,sb,aC));

[place holder]

Cyclical permutations of each of the foregoing functions, as well.

Strategy

This strategy is specifically for the solution of spherical triangles (but, may be employed for plane triangles, as well), employing the foregoing functions. We adapt the reasonable, but purely arbitrary, convention that we will view a sphere from the outside and that we will traverse the boundary of a region on the sphere so that the region is on our left-side. There are two difficulties:

While clearly left and right are distinct, there is no non-anthropomorphic way of defining "left".
What should one do with self-intersecting figures, like an eight?

With the foregoing convention, we distinguish the triangles ABC and ACB.

The ultimate goal is the solution of the triangle. That is, to find the three sides {sa,sb,sc} and the opposite three angles {aA,aB,aC}. We partition this problem according as the emphasis is on the sides or the angles, with the latter being the dual of the former.

sides.

We take as a sub-goal that of obtaining the three sides and only the one angle.

Given two sides and an angle opposite one of them SSA. There are two non-equivalent solutions. Employ the short form of the law of sines - law_sin_short_aA --, to obtain the angle opposite of the other side. Now, we have:
Given two sides and the angles opposite each of them ASSA. Employ the Napier analogy one - napier_one_sc --, to obtain the side between the two known angles. Now, we have:
Given the three sides and two of the angles SASAS. It is tempting to employ the short form of the law of sines - law_sin_short_aA --, to obtain the remaining angle. Thus, finishing the solution. However, because of the ambiguities inherent in each of the functions, this solution may not be self-consistent. It is more prudent to discard (temporarily) one of the angles and have:
sub-goal. Given the three sides and one of the angles SASS. Employ the combined haversine and sine (short form) laws - law_hav_and_sin_aB --, to obtain the second angle. Then, permute this function cyclically, to obtain the third angle. Now, we have obtained a self-consistent solution.
Given the three sides SSS. Employ the law of haversines - law_hav_aA --, to obtain one of the angles. Now, we have the sub-goal case.
Given two sides and the included angle SAS Employ the law of haversines - law_hav_sa --, to obtain the remaining side, i.e., the side opposite the given angle. Now, we have obtained the sub-goal case, again.

[place holder]

angles.

The angles are the dual of the sides, above.

We take as a sub-goal that of obtaining the three angles and only the one side.

Given two angles and a side opposite one of them AAS. In the spherical case, there are two non-equivalent solutions. Employ the dual of the short form of the law of sines - law_sin_short_sa --, to obtain the side opposite the other angle. Now, we have:
Given two angles and the sides opposite each of them SAAS. Employ the dual of the Napier analogy one - napier_one_aC --, to obtain the angle between the two known sides. Now, we have:
Given the three angles and two of the sides ASASA. It is tempting to employ the short form of the dual of the law of sines - law_sin_short_sa --, to obtain the remaining side. Thus, finishing the solution. However, because of the ambiguities inherent in each of the functions, this solution may not be self-consistent. It is more prudent to discard (temporarily) one of the sides and have:
sub-goal. Given the three angles and one of the sides ASAA. Employ the dual of the combined haversine and sine (short form) laws - law_hav_and_sin_sb --, to obtain the second side. Then, permute this function cyclically, to obtain the third side. Now, we have obtained a self-consistent solution.
Given the three angles AAA. (This case cannot be solved in the plane; because, the dual of the law of haversines is not available therein). Employ the dual of the law of haversines - law_havdual_sa --, to obtain one of the sides. Now, we have the sub-goal case.
Given two angles and the included side ASA. Employ the dual of the law of haversines - law_havdual_aA --, to obtain the remaining angle, i.e., the angle opposite the given side. Now, we have obtained the sub-goal case, again.

[place holder]

plane angles.

This is easier than the foregoing "angles", which is applicable to either the spherical or plane cases.

Given three angles AAA. Cannot be solved in the plane-case.
Given two angles and a side. Either given two angles and a side opposite one of them AAS xor given two angles and the included side ASA. Employ the dual of the law of cosines -- law_cosdual_plane_aA --, to obtain the third angle. Now, we have the
Given three angles and a side opposite one of them ASAA. Employ the short form of the law of sines-- law_sin_short_plane_sa --, cyclically, to find the remaining sides. Now, we have solved the triangle.

[place holder]

alternatives.

Instead of the Napier analogy one, one may employ the corresponding Napier analogy two. Instead of the law of haversines, one may employ the corresponding law of half-tangents. Except for the resulting large errors, the corresponding law of cosines could be employed, as well.

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