To make the equations clearer, we omit any punctuation at their ends.

We presuppose a familiarity with the concepts of **points**
and **vectors**. The trigonometric functions are
convenient combinations of the exponential
function. We already have discussed both the circular and the hyperbolic trigonometric functions.

Tacitly, we assume that the geometry is a plane geometry (curvature equal to zero). Later we will show the generalization to the two remaining elementary geometries, namely, a spherical geometry (strictly positive curvature) and a hyperbolic geometry (strictly negative curvature).

The **cross product** (sometimes called the
vector product) of two vectors is a **pseudo-vector**,
and as such, is dependent upon the **chirallity** of
the space. The cross product cannot be defined in less than
three-dimensional space. In the mid 1920’s, **Cartan**
generalized the cross product to higher than three-dimensions.
However, this generalization is too complicated to be practical.
The same functionality may be achieved easily by the use of
matrices. Thus since the cross product is practical only in
three-dimensions, we will not consider the cross product at all.

Observe that the cross product of a pseudo-vector by a vector is, once again, a vector.

However, one must not underestimate the utility of the cross product. Maxwell’s electromagnetic theory, either in its elegant Hamilton’s quaternion or vulgar Gibb’s vector form, expresses the magnetic pseudo-vector as a cross product of the electric and velocity vectors. And closes the relationships by expressing the electric vector as the cross product of the magnetic pseudo-vector and current vector. Each of these is a first-order differential equation.

The Einstein’s relativity theory owes its difficulty to its avoidance of the cross product. By eliminating the magnetic pseudo-vector, we obtain a second-order differential equation.

As the only product of vectors that we consider, we will omit
the dot in the dot product. The **dot product**
(sometimes called the scalar product) of two vectors (emanating
from a point) is a scalar, equal to the product of the norms of
each of these two vectors times the cosine of the included angle.
We turn this definition around to define the angle, in the closed
interval [0, pi]. Then the norm of a vector (which is written
||vector||) is equal to the square root of its square; that is of
the dot product of the vector by itself.

Given three vectors a, b, and c, the equation of a **triangle**
is

a + b + c = 0

Solve for a and square to obtain

a^2 = b^2 + c^2 – 2 (-b) c

Evaluation of this dot product yields the first version of the
**Law of Cosine**

a^2 = b^2 + c^2 – 2 ||b|| ||c|| cosine(A) = (||b|| – ||c||)^2 + 2 ||b|| ||c|| versine(A)

where the capital letter indicates the angle opposite the like-named vector. Cyclical permutation yields the remaining two equations. Then, solve for the angle to obtain the second version of the Law of Cosine

cosine(A) = ((b^2 + c^2) – a^2) / (2 ||b||
||c||))

versine(A) = (||a|| – (||b|| – ||c||)) (||a|| + (||b||
– ||c||)) / (2 ||b|| ||c||))

Again, cyclical permutation yields the remaining two equations. Despite their popularity, because of their excessive round-off errors; these equations involving the cosine should not be employed for numerical calculation. The equations involving the obscure function, versine, however, circumvent the round off problem. Indeed, these equations are the motivation for the original definition of the versine function. I do not know any application for the coversine function. Probably, its only reason for being is as the co-function to the versine function.

Definition: A **circle** is the locus of point at
the end of vectors, whose norm is the radius from the point of
the center.

Definitions: A **circumscribing circle** passes
through the vertices of the triangle. An inscribed circle passes
through the foot of a perpendicular dropped from the center to
each of the sides of the triangle.

Square the foregoing equation

(cosine(A))^2 = a^2 ((a^4 + b^4 + c^4) - 2 (b^2 c^2 + c^2 a^2 + a^2 b^2)) / (4 a^2 b^2 c^2) + 1

Then, by use of the identity

1 = (sine(A))^2 + (cosine(A))^2

we obtain

(sine(A))^2 = a^2 (2 (b^2 c^2 + c^2 a^2 + a^2 b^2) – ( a^4 + b^4 + c^4)) / (4 a^2 b^2 c^2)

And taking square roots, we obtain the **Law of Sine**

sine(A) / ||a|| = 1 / (2 R)

where R, the radius of the circumscribed circle, is the positive square root of

R^2 = (a^2 b^2 c^2) / (4 (b^2 c^2 + c^2 a^2 + a^2 b^2) – (a^2 + b^2 + c^2)^2)

As it should, R is a symmetric form in the three vectors. That
R indeed it the radius of the circumscribed circle will be shown
later. By cyclic permutation of the Law of Sine, we obtain the **Law of Sines**

2 R = ||a|| / sine(A) = ||b|| / sine(B) = ||c|| / sine(C)

In this form, the Law of Sines may be employed to solve a
triangle whose three sides (**SSS**) are known.
Alternatively, the Law of Tangents may be employed for the SSS.
If either two sides and an angle opposite one of them (**SSA**)
or two angles and a side opposite one of them (**AAS**)
is know, the Law of Sines yields the remaining element of the
other pair. In either case, we have obtained two sides and the
angles opposite them. The third angle follows from the equation

pi + area / rho^2 = A + B + C

which we will prove later. Now that we have all three angles and at least one side, the remaining sides may be obtained from the Law of Sines.

The only situation, which the Law of Sines does not solve, is that of two sides and the included angle (SAS). Most people succumb to the temptation of the Law of Cosine for this purpose. Instead, one should employ its variant, the Law of Versine.

The final situation of two (**AA**) or of three
angles (**AAA**) yields only a similar triangle;
because we cannot find any of the sides. In a non-flat space (rho
not equal to zero), however, we can solve the AAA problem.

There are constraints upon the sides and angles, for instance, the sum of the norms of the two sides with the smaller norms has to exceed the norm of the remaining side. Other constraints are left for the reader to discover. The SSA, with the angle being acute, has two solutions; or none, if the applicable constraint is not satisfied.

**Theorem**: **Pythagorean**. The
sum of the squares of the sides (called legs) including a right
angle is equal to the square of the other side (called
hypotenuse), opposite the right angle.

Proof: Set the angle in the Law of Cosine equal to pi / 2. Q.E.D.

Note: This is the trigonometric statement of the Pythagorean theorem; the geometric has “square(s) on”.

**Theorem**: The sum of the exterior angles of a
polygon is an integral (dependent upon the winding) multiple of
two-pi minus the quotient of enclosed area divided by the radius
of curvature, rho, squared. Each of the formulae is written with
the assumption that the winding is non-negative.

Proof follows from the modulus of periodicity of the sine and cosine functions. Q.E.D.

**Corollary**: The sum of the interior angles of
a triangle is pi minus the quotient of its enclosed area divided
by the radius of curvature, rho, squared.

Proof: Obviously the winding of a triangle can be only either plus or minus one. Let the angles of a triangle be A, B, and C. Then the exterior angles are pi – A, pi – B, and pi – C. Substitute them into the foregoing theorem (pi – A) + (pi – B) + (pi – C) = 2 pi – area / rho^2. Rearrange to yield A + B + C = pi + area / rho^2. Q.E.D.

This equation may be solved for the radius of curvature, rho, to yield

rho^2 = area / (A + B + C – pi)

**Definition**: Two triangles are said to be **similar**
iff (= if and only if) each of their respective angles is equal
to each other.

**Definition**: Two triangles are said to be **congruent**
iff (= if and only if) they are similar and each of their
respective sides is equal to each other.

**Theorem: SSS**. Two triangles are congruent if
each of their respective sides is equal to each other.

Proof follows from the Law of Cosine. Q.E.D.

**Theorem**: The R of the Law of Sines indeed is
the radius of the circumscribing circle.

Proof: Solve the original triangle and call its sides a, b, and c and the angles opposite A, B, and C, respectively. Take a point P and draw a circle of radius R with P as its center. Take the central angels 2 A, 2 B, and 2 C, whose sum, of necessity, is two pi. By the Law of Sines, the lengths of the chords are a, b, and c, respectively. By SSS, this triangle is congruent to the original triangle. Q.E.D.

**Corollary**: Each central angle of the
circumscribing circle is twice the corresponding vertical angle
of the triangle.

Proof was demonstrated in the foregoing construction. Q.E.D.

**Corollary**: The angle between two lines, which
intersect in the interior (exterior) of a circle, is equal to
one-half of the sum (difference, respectively) of the central
angles.

Proofs are obvious from the previous corollary. Q.E.D.

**Corollary**: The hypotenuse of a right triangle
is a diagonal of the circumscribing circle.

Proof is obvious from the penultimate corollary. Q.E.D.

**Theorem**: Two similar triangles have their
respective sides proportional to each other and proportional to
the respective radius of the circumscribing circles.

Proof follows from the Law of Sines. Q.E.D.

**Axioms**: The **metric** of our
space is **homogeneous**, **orthogonal**,
and **isotropic**. At first, we tacitly assume that
the radius of curvature, rho, is zero; that is, that the space is
a plane. Presently, we will show that the only modification
required is that the norm of a vector be the radius of curvature
times the absolute value of the sine of the angle from the center
of curvature.

**Definition**: A pair of lines is said to be **parallel**,
if there exists a transverse, which is perpendicular to each of
the given lines.

Observe that the winding of a **quadrilateral**
is at most one, in magnitude.

**Definition**: A **trapezoid** is a
quadrilateral, of non-zero winding, with one pair of
opposite-sides parallel.

**Definition**: A** parallelogram**
is a trapezoid with the other par of opposite-sides parallel,
also.

**Theorems**: Alternate angles of a parallelogram
are equal. Opposite sides of a parallelogram are equal, in
magnitude. The diagonals of a parallelogram bisect each other.

Proof: Construct a diagonal, to partition the parallelogram into two triangles, which are congruent. Q.E.D.

**Definition**: A **rectangle** is a
parallelogram, with one right angle.

**Corollary**: The other three angles also are
right angles.

Proof follows from the penultimate theorem.

**Definition**: A **rhombus** is a
parallelogram, with one pair of adjacent-sides equal, in norm.

**Theorem**: The diagonals of a rhombus are
perpendicular bisectors of each other.

Proof: Construct a diagonal, to partition the rhombus into two triangles, which are congruent. Q.E.D.

**Definitions**: A **square** is a
rhombus with one right angle. A **square** is a
rectangle with one pair of adjacent sides equal, in norm.

**Corollary**: These two definitions are
equivalent.

Proofs are obvious. Q.E.D.

From the theory of **measure**, we know that: (1)
The area of a rectangle is the product of the norms of a pair of
adjacent sides. This statement is true only in the differential
sense, if the curvature of the space is non-zero. (2) The area of
a partitioned region is equal to the sum of the areas of its
partitions.

**Definition**: In a triangle, the line from a
vertex perpendicular to the side opposite, called the base, is
called the corresponding altitude.

**Definition**: The **cardinality**
of a set is the collection of sets, which may be placed in
one-to-one correspondence with the given set. Any one of these
sets may be singled out as the representative.

**Definition**: The **geometric mean**
is defined as the exponential of the arithmetic mean (provided
that it exists) of the logarithms of a set of numbers.

**Corollary**: If the cardinality of this set is
two, then the geometric mean is the square root of the product of
the two numbers.

Proof: This is an obvious special case. Q.E.D.

**Theorem**: The altitude to its hypotenuse
partitions a right triangle into two similar triangles, similar
to the original triangle.

**Corollary**: In a right triangle, the norm of
the altitude to its hypotenuse is the geometric mean of the norms
of the segments of the hypotenuse, into which this altitude has
partitioned its base.

**Lemma**: The area of a parallelogram is the
product of the norm of a base and the norm of the corresponding
altitude. This statement is true only in the differential sense,
if the curvature of the space is non-zero.

Proof: Partition off a triangular end and place it at the opposite end. Q.E.D.

**Theorem:** The area of a triangle is equal to
one-half of the product of the norm of a base and the norm of the
corresponding altitude. This statement is true only in the
differential sense, if the curvature of the space is non-zero.

Proof: Consider the triangle as obtained by the partitioning, into a congruent pair of triangles, of a parallelogram by one of its diagonals. Q.E.D.

**Corollary**: The area of a triangle is one
fourth of the product of the norms its sides divided by the
radius of the circumscribed circle: ||a|| ||b|| ||c|| / (4 R)
This formula is true only in the differential sense, if the
curvature of the space is non-zero. This formula is, as it should
be, a symmetric function of the sides.

Proof follows from the Law of Sine. Q.E.D.

**Theorem**: The area of a triangle is equal to
one-half of the product of the radius of the inscribed circle and
the sum of the norms of the sides. r (||a|| + ||b|| + ||c||) / 2
This formula is true only in the differential sense, if the
curvature of the space is non-zero.

Proof is obvious from the definition of an inscribed circle. Q.E.D.

**Theorem**: The radius of the inscribed circle
is

r = ||a|| ||b|| ||c|| / (8 R (||a|| + ||b|| + ||c||))

Proof: Equate the preceding two formulae for the area of a triangle. Q.E.D.

We have obtained the Euclidean plane geometry, easily and concisely and without drawing any pictures. In this case, trigonometry is worth a thousand pictures.

At present, the proofs are omitted. Even so, we do better than most recent textbooks, which even do not mention any of these laws.

For brevity, let s = (||a|| + ||b|| + ||c||) / 2

There is a set of equations, known as **de’Molwede’s
(sp?) Formulae**; but this textbook does not list them.

(||b|| – ||c||) / (||b|| + ||c||) = tangent((B – C) / 2) / tangent((B + C) / 2)

Proof: For brevity of notation, we omit the magnitude bars. Write the Law of Sines as

a / b = sin(A) / sin(B)

and add or subtract one to each side to obtain, respectively,

(a + b) = (sin(A) + sin(B)) / sin(B)

(a - b) = (sin(A) - sin(B)) / sin(B)

Then, by the use of the Sums or Differences, the quotient is

(a + b) / (a - b) = 2 sin((A + B) / 2) cos((A - B) / 2) / (2 cos((A + B) / 2) sin((A - B) / 2))

which,, by the definition of the tangent, simplifies to the Law of Tangents. QED.

tangent(A / 2) = r / (s – ||a||)

The area of a triangle is area^2 = s (s –
||a||) (s – ||b||) (s – ||c||)

r^2 = (s – ||a||) (s – ||b||) (s – ||c||) / s

R = ||a|| ||b|| ||c||| / (4 area)

R^2 = (||a|| ||b|| ||c||)^2 / (16 s (s – ||a||) (s –
||b||) (s – ||c||))

Since the sides of a triangle are great circles, we express them in terms of the sine of the angle, measured from the center of curvature. Thus, we observe that in the Law of Sines, the norm of a side becomes the sine.

Elementary textbooks confuse the issue by a multitude of drawings. It takes a thousand words to dispel the confusion caused by one picture. Advanced textbooks assume that the reader knows how to derive these formulae. Thus their formal derivation never is shown, except here.

For generality and easier verification of consistency, as we already did for the plane triangle, we begin with the dot product. Consider a sphere, centered on the origin. Let u, v, w be three non-coplanar unit-vectors from the origin. Their extensions will intersect the sphere at three points.

Since, by definition, these vectors have a unit norm, u . u = v . v = w . w = 1. Define the angles a, b, c in the open interval (0, pi) by the implicit equations

- v . w = cos(a)
- w . u = cos(b)
- u . v = cos(c)

Then R = v - v . u u is in the u-v plane. R . u = 0 proves that R is normal to u.

Since the norm of R is sin(c), the unit vectors are

- R / || R || = (v - u cos(c)) / sin(c)
- S / || S || = (w - u cos(b)) / sin(b)

We have their dot product

cos(A) = (cos(a) = cos(c) cos(b)) / (sin(c) sin(b)),

where A, B, C are in the open interval (0, pi). Then
rearranging, we obtain the **Law of Cosines**

cos(a) = sin(b) sin(c) cos(A) + cos(b) cos(c).

Rearange once again

sin(b) sin(c) cos(A) = cos(a) - cos(b) cos(c)

Square and replace each square of a cosine, using its identity,

- (sin(b) sin(c) sin(A))^2 = 2 - ((sin(a)))^2 + (sin(b))^2 + (sin(c))^2) - 2 cos(a) cos(b) cos(c).

Divide by (sin(a) sin(b) sin(c))^2 and take its square root,
to obtain the **Law of Sines**

sin(A) / sin(a) = sin(B) / sin(b) = sin(C) / sin(c) = a symmetric function in a, b, c.

The Law of Cosines may be written in terms of the haversine as
the **Law of Haversines**

haversine(a) = haversine(b – c) + sine(b) sine(c) haversine(A)

If we consider u, v, w to be the vertices; then a, b, c are the sides and A, B, C are the angles of the spherical triangle. On the other hand, if we consider u, v, w to be the normals to the planes; then a, b, c are the sides and A, B, C are the angles.

For brevity, let s – (a + b + c) / 2 and sigma = (A + B + C) / 2

The remaining equations have to be derived from these two
laws; but it is a bear to do so.

There is a set of equations, known as **de’Molwede’s
(sp?) Formulae**; but this textbook does not list them.

cosine(c) = cosine(a) cosine(b) = cotangent(A)
cotangent(B)

sine(a) = sine(A) sine(c)

tangent(a) = cosine(B) tangent(c)

tangent(a) = tangent(A) sine(b)

cosine(A) = cosine(a) sine(B)

As was true of the plane counterpart, except for the one involving the haversine, these equations suffer from excessive round-off errors. Again, this equation is the motivation for the original definition of the haversine function.

(tangent(r))^2 = sine(s – a) sine(s –
b) sine(s – c) / sine(s)

tangent(A / 2) = tangent(r) / sine(s – a) = tangent(rho)
cosine(sigma - A)

(tangent(rho))^2 = - cosine(sigma) / (cosine(sigma – A)
cosine(sigma – B) cosine(sigma – C))

tangent((a – b) / 2) / tangent(c / 2) =
sine((A – B) / 2) / sine((A + B) / 2)

tangent((a + b) / 2) / tangent(c / 2) = cosine((A – B) / 2)
cosine((A + B) / 2)

tangent((A – B) / 2) / cotangent(C / 2) = sine((a – b)
/ 2) / sine((a + b) / 2)

tangent((A + B) / 2) / cotangent(C / 2) = cosine((a – b) /
2) / cosine((a + b) / 2)

Definition: The spherical excess is the sum of the internal angles of a triangle, less pi

E = A + B + C – pi

Then

E = area / rho^2

(tangent(E / 4))^2 = tangent(s / 2) tangent((s – a) / 2)
tangent((s – b) / 2) tangent((s – c) / 2)

As a check, the limit, as the curvature rho approaches zero, of any spherical equation is a valid plane-equation. For instance, the foregoing equation becomes Heron’s formula for the area of a plane triangle. On the other hand, a strictly negative curvature yields a hyperbolic space.

A pair of** intersecting** curves has one point
in common. A pair of **tangent** curves has to
coincident points in common. A pair of **osculating**
curves has three coincident points in common.

At a given point of a given curve, construct the osculating
circle. The center of this circle is the **center of
curvature**. The radius of this circle is the **radius
of curvature**, whose reciprocal is the **curvature**.
The radius of curvature often it taken as a vector, emanating at
the center of curvature, towards the given point on the curve.
Then, the loci of these centers of curvature constitute the **involute**,
with the original curve being called the **evolute**.
A variant on this concept employs a circle of fixed radius, for
instance, in the study of the path of a reamer or of a cam.

The cross product of two vectors is defined as a pseudo-vector, with the following properties: (1) Its magnitude is the product of their norms multiplied by the sine of the included angle. (2) Its direction is normal to both vectors. (3) Its chirallity is the same as that of the space.

The **cross product** is **skew-symmetric**

b x a = – (a x b)

The cross product of a vector and a pseudo-vector is, back again, a vector

a x (b x c) = b (a c) – c (a b)

A curve has the **local coordinates**: (1) The **unit
tangent**; i.e., a unit vector in the direction of the
velocity. (2) The **normal**, which is a unit vector
towards the center of curvature. (3) The **binormal,**
which is a unit vector in the direction of the cross product of
the preceding two vectors.

**Torsion** is defined as the rate of change of
the projection on the normal-binormal plane of the direction of
the normal, with respect to arc length.

Embed a surface of one or more dimensions in a flat space of minimal dimensions. For example, consider an ellipsoid, embedded in a flat space of one higher dimension.

A** geodesic** locally is the shortest path.

Construct a family of geodesic curves drawn through a given
point on a surface. Along the tangent of each curve, construct a
vector of length equal to the curvature of that curve. The result
is an **ellipsoid of curvature**.

Solve the eigenvalue-eigenvector problem. The eigenvectors are
the principal directions, which may be shown to be mutually
orthogonal. By convention, the eigenvectors are taken to have a
unit norm. The eigenvalues are the principal curvatures. Their
respective products are the principal axes of the ellipsoid of
curvature. The geometric mean of these eigenvalues is the
curvature of the surface, rho. This is the formal definition of
the **curvature**, which we have been employing
throughout the previous discussion of Elementary Geometry.

Alternatively, along the tangent of each curve, construct a
vector of length equal to the norm of the radius of curvature.
The eigenvectors will have the same directions. The eigenvalues
are the principal radii of curvature. Their respective products
are the principal axes of the **ellipsoid of radii of
curvature**. This ellipsoid osculates the given surface
and coincides with the given surface if it had been an ellipsoid.

If the original surface had been a sphere, all of the eigenvalues would have been the same and equal to the curvature or radius of curvature, respectively, of the surface. The radius of curvature of a sphere is equal to its radius. The most common example is a two-dimensional sphere embedded in three-dimensional flat space.

Each principal direction may be either **spherical**
(positive curvature), **flat** (zero curvature), or **hyperbolic**
(negative curvature). These individual curvatures need not have
the same sign. However, I do not know what implications a **mixed-sign
curvature** has upon the space curvature rho, the
spherical excess E, and the area enclosed by a closed curve (a
triangle, for instance). If anybody knows and explains to me (or
can provide me with a reference to an in-print English-language
textbook, which answers this question), I will be grateful. **Algebraic
topology**, employing **denumerable groups**,
deals with the structure of spaces which are not **simply
connected**.

**Curiously**, the **spelling checker**
likes neither binormal, chirallity, conversine, denumerable,
eigenvalue, eigenvector evolute, haversine, isotropic, modulus,
orthogonal, orthonormal, nor versine.

Copywrite © 1997 R. I. 'Scibor-Marchochi last modified Monday 01-st of September 1997.