Angles in analytic geometry

David Pierce

November 3, 2006; recompiled, November 9, 2007

The words synthetic and analytic are sometimes used as opposites or complements. The geometry pioneered by Rene Descartes [1] is called analytic geometry; by contrast, the geometry of ancient mathematicians like Euclid of Alexandria [2] and Apollonius of Perge [3] is called synthetic geometry.

The word synthetic comes from the Greek ���������� meaning skilled in putting together or constructive. This Greek adjective derives from the verb ��������� put together, construct. The word analytic is the English form of ����������, which derives from the verb ������ undo, set free, dissolve.

What do these words mean in the context of mathematics? Although we refer to ancient geometry as synthetic, the Ancients evidently recognized both analytic and synthetic methods. Pappus of Alexandria writes:

Now analysis is a method of taking that which is sought as though it were admitted and passing from it through its consequences in order to something which is admitted as a result of synthesis; for in analysis we suppose that which is sought to be already done, and we inquire what it is from which this comes about, and again what is the antecedent cause of the latter, and so on until, by retracing our steps, we light upon something already known or ranking as a first principle; and such a method we call analysis, as being a reverse solution.

But in synthesis, proceeding in the opposite way, we suppose to be already done that which was last reached in the analysis, and arranging in their natural order as consequents what were formerly antecedents and linking them one with another, we finally arrive at the construction of what was sought; and this we call synthesis. [4, p. 597]

The main point seems to be that synthesis (and synthetic geometry in particular) should start from first principles and build from there; while analysis (and analytic geometry) is a kind of search for principles from which a desired result would follow.

Euclid of Alexandria begins his Elements with five principles:

1. any two points can be joined by a [straight] line;

2. any [straight] line can be extended indefinitely; 3. a circle can be drawn with any center and radius;

4. all right angles are equal;

5. if two angles, say ABC and BCD, are together less than two right angles, then lines BA and CD, extended as necessary beyond B and D, must meet.

The 47th proposition that Euclid derives from these principles is commonly known by another name: Theorem 1 (Pythagoras). In a right triangle, the square on the hypotenuse is equal to the squares on the legs.

A B  C D  E F G  H  K LM Proof. The proof is based on the picture at

the right, where ABC is a triangle with right angle at A, the squares on the sides are drawn as shown, and AL is perpendicular to BC.

The square ABF G is twice the triangle CBF, which is congruent to DBA, which is half the rectangle DBML. So ABF G is equal to DBML. Likewise, ACKH is equal to ECML. But DBML and ECML are together the square BCED.

One way to analyze the Pythagorean Theorem is to understand it as ‘really’ being about lengths: If the side of ABC opposite angle A has length a, and so forth, and the angle at A is right, then

a^{2} = b^{2} + c^{2}.  (1)

We shall see how such an equation can arise when we understand the points A, B, and C as ordered pairs (or triples) of real numbers.

O  A B D P C  In Euclidean geometry, two distinct lines intersecting

at a point determine a plane in the following sense. Let the lines be OA and OB. If C is on OA, and D is on OB, then there is a unique parallelogram CODP for some point P. (The parallelogram is ‘degenerate’ if C or D is O.) Such points P compose a plane, and every point P in this plane determines a unique such parallelogram. Therefore, instead of working with the points P, we can work with the pairs (x, y), where x is the ‘signed’ distance of C from O (that

is, x is negative if it is on the opposite side of OA from A), and y is the signed distance of D from O. Here we understand signed distances to be just real numbers; so our plane becomes the set ℝ × ℝ or ℝ^{2} of ordered pairs of real numbers.

So plane analytic geometry is about the set ℝ^{2}; we think of its elements as points. We conceive of ℝ^{2} as having axes, called X and Y respectively. The X-axis consists of points (x, 0); the Y -axis consists of points (0, y). Nothing that we have said so far requires these axes to be perpendicular; indeed, it is not yet clear what it would mean for the axes to be perpendicular, since these axes are just sets of ordered pairs of numbers. However, Equation (1) is a clue.

X Y (a, a^{2}) (0, −a^{2}) (x, 2ax − a^{2}) (x, x^{2}) Not everything interesting that we can say about ℝ^{2} requires us to conceive of the axes as perpendicular. For example, from the inequality

0 ⩽ (x − a)^{2} = x^{2} − 2ax + a^{2},

we obtain

2ax − a^{2} ⩽ x^{2}.

This means that every point on the curve defined by y = x^{2} is above the point with the same X-coordinate on the line y = 2ax−a^{2}. As the picture shows, this makes visual sense, even if the two axes are not perpendicular.

The same element of ℝ^{2} can be written as u⃗ or (u_{1}, u_{2}). Then u_{1}^{2} + u_{2}^{2} ⩾ 0, so \sqrt{u_{1}^{2} + u_{2}^{2}} is a well-defined, non-negative number: let us call this number the norm of u⃗ and denote it by

|u⃗|.

So, by definition, we have the identity

|u⃗| = \sqrt{u_{1}^{2} + u_{2}^{2}}.  (2)

The norm is intended to express a notion of distance: |u⃗| should make sense as the distance between u⃗ and 0 (that is, (0, 0)). Does it make sense? Well, in Euclidean geometry, u⃗ is the length of the hypotenuse of a right triangle whose legs have lengths u_{1} and u_{2}. But what is a right triangle in ℝ^{2}?

We can add elements of ℝ^{2} coordinate-wise:

u⃗ + v⃗ = (u_{1} + v_{1}, u_{2} + v_{2}).  (3)

Likewise, we can multiply them by real numbers:

a · u⃗ = (a · u_{1}, a · u_{2}).  (4)

Here a may be called a scalar; the elements of ℝ^{2} are then called vectors. The operations on vectors have various nice properties that follow from the corresponding properties of operations on scalars.

Two vectors are parallel if one of them is a scalar multiple of the other: if a · u⃗ = v⃗, or a · v⃗ = u⃗, then

u⃗ ∥ v⃗.  (5)

Some algebraic consequences of (2) follow almost immediately: Theorem 2. For all u⃗ in ℝ^{2} and a in ℝ,

1. 0 ⩽ |u⃗|;

2. 0 = |u⃗| ⟺ 0 = u⃗;

3. |a · u⃗| = |a| · |u⃗|.

Proof. We observed (1) while defining |u⃗|. For (2), the direction ⇐ follows by computation: |0| = \sqrt{0^{2} + 0^{2}} = 0; for the direction ⇒, we prove the contrapositive: If 0 ≠ u⃗, then one of u_{1} and u_{2} is not 0; without loss of generality, we may assume u_{1} ≠ 0. Then

|u⃗|^{2} = u_{1}^{2} + u_{2}^{2} ⩾ u_{1}^{2} > 0,

so |u⃗| > 0, and in particular |u⃗| ≠ 0. Finally, for (3), just compute:

|a · u⃗| = |(a · u_{1}, a · u_{2})|
= \sqrt{(a · u_{1})^{2} + (a · u_{2})^{2}}
= \sqrt{a^{2} · (u_{1}^{2} + u_{2}^{2})}
= |a| · \sqrt{u_{1}^{2} + u_{2}^{2}},

which is |a| · |u⃗|.

The theorem does not violate any notion of |u⃗| as the distance between u⃗ and 0. For example, if a · u⃗ = v⃗, then the distance from 0 to v⃗ ought to be |a| times the distance to u⃗; but this is what (3) expresses.

But we should like |u⃗ + v⃗| to make sense as the length of the side of a triangle whose other two sides have lengths |u⃗| and |v⃗|. In particular, we want

|u⃗ + v⃗| ⩽ |u⃗| + |v⃗|.  (6)

We cannot assume that this is true; it is already either true or not, since it is stating a possible property of ℝ. In fact, we shall prove that it is true. Towards doing so, we note first that (since norms are non-negative,) (6) is logically equivalent to

|u⃗ + v⃗|^{2} ⩽ (|u⃗| + |v⃗|)^{2}
= |u⃗|^{2} + 2 · |u⃗| · |v⃗| + |v⃗|^{2},

which is equivalent to

\frac{|u⃗ + v⃗|^{2} − |u⃗|^{2} − |v⃗|^{2}}{2} ⩽ |u⃗| · |v⃗|.  (7)

It turns out to be convenient to give the left-hand member of this inequality an abbreviation and a name: it is the scalar product or dot-product of u⃗ and v⃗, and it is denoted

u⃗ · v⃗.

So, by definition, we have the identity

u⃗ · v⃗ = \frac{|u⃗ + v⃗|^{2} − |u⃗|^{2} − |v⃗|^{2}}{2} .  (8)

Presently we shall see an alternative expression for u⃗ · v⃗; but let us first note that some basic properties of the scalar product follow directly from (8): Theorem 3. For all u⃗ and v⃗ in ℝ^{2}:

1. u⃗ · v⃗ = v⃗ · u⃗; 2. u⃗ · 0 = 0;

3. u⃗ · u⃗ = |u⃗|^{2}. Proof. Left to the reader. To be able to say much more, we need: Lemma 4. For all u⃗ and v⃗ in ℝ^{2},

u⃗ · v⃗ = u_{1} · v_{1} + u_{2} · v_{2}.  (9)

Proof. Just compute. The lemma allows us to show: Theorem 5. For all u⃗, v⃗, and w⃗ in ℝ^{2}, and a in ℝ,

1. u⃗ · (v⃗ + w⃗) = u⃗ · v⃗ + u⃗ · w⃗;

2. u⃗ · (a · v⃗) = a · (u⃗ · v⃗). Proof. Computation. A special case of (2) is

u⃗ · (−v⃗) = −u⃗ · v⃗.

Using this, in (8), we can replace v⃗ with −v⃗ and rearrange to get

|u⃗ − v⃗|^{2} = |u⃗|^{2} + |v⃗|^{2} − 2 · u⃗ · v⃗.  (10)

Note the similarity to the Law of Cosines. Theorem 6 (Cauchy–Schwartz). For all u⃗ and v⃗ in ℝ^{2},

|u⃗ · v⃗| ⩽ |u⃗| · |v⃗|,  (11)

with equality if and only if u⃗ ∥ v⃗. Proof. Let x be a scalar. Now matter how x changes, we must have

0 ⩽ |u⃗ − x · v⃗|,

equivalently,

0 ⩽ |u⃗ − x · v⃗|^{2}.  (12)

Now compute:

|u⃗ − x · v⃗|^{2} = (u⃗ − x · v⃗) · (u⃗ − x · v⃗)
= u⃗ · u⃗ − 2x · u⃗ · v⃗ + x^{2} · v⃗ · v⃗  [by Theorem 5]
= |u⃗|^{2} − 2x · u⃗ · v⃗ + x^{2} · |v⃗|^{2}  [by Theorem 3]

This is a quadratic polynomial in x; it may be written in the more usual fashion as

|v⃗|^{2} · x^{2} − 2 · (u⃗ · v⃗) · x + |u⃗|^{2}.  (13)

By the general theory of such things, the polynomial ax^{2}+bx+c takes an extreme value at −b/(2a), and this extreme value is c−b^{2}/(4a); this is a maximum value if a > 0. In particular, our polynomial (13) has minimum value

|u⃗|^{2} − \frac{(u⃗ · v⃗)^{2}}{|v⃗|^{2}} .

This cannot be negative, by (12). That is,

0 ⩽ |u⃗|^{2} − \frac{(u⃗ · v⃗)^{2}}{|v⃗|^{2}} ,

\frac{(u⃗ · v⃗)^{2}}{|v⃗|^{2}} ⩽ |u⃗|^{2},

(u⃗ · v⃗)^{2} ⩽ |u⃗|^{2} · |v⃗|^{2},

and therefore

|u⃗ · v⃗| ⩽ |u⃗| · |v⃗|.

Finally, this inequality is an equation if and only if it is possible for u⃗ − x · v⃗ to be 0; but this is possible if and only if u⃗ and v⃗ are parallel.

If we accept that there is a function cos on ℝ that takes on every value in the interval [−1, 1], then, by the Cauchy–Schwartz Inequality (11), Theorem, if u⃗ and v⃗ are non-zero, there is θ such that

cos θ = \frac{u⃗ · v⃗}{|u⃗| · |v⃗|};  (14)

in particular, cos θ = 1 if and only if u⃗ ∥ v⃗. Rewriting (14) as

|u⃗| · |v⃗| · cos θ = u⃗ · v⃗  (15)

and substituting into (10), we get a more familiar form of the Law of Cosines.

References

[1] René Descartes. The Geometry of René Descartes. Dover Publications, Inc., New York, 1954. Translated from the French and Latin by David Eugene Smith and Marcia L. Latham, with a facsimile of the first edition.

[2] Euclid. The thirteen books of Euclid’s Elements translated from the text of Heiberg. Vol. I: Introduction and Books I, II. Vol. II: Books III–IX. Vol. III: Books X–XIII and Appendix. Dover Publications Inc., New York, 1956. Translated with introduction and commentary by Thomas L. Heath, 2nd ed.

[3] Apollonius of Perga. On Conic Sections, Books I–III. Great Books of the Western World. Encyclopædia Britannica, 1952. Translated by R. Catesby Taliaferro.

[4] Ivor Thomas, editor. Selections illustrating the history of Greek mathematics. Vol. II. From Aristarchus to Pappus. Harvard University Press, Cambridge, Mass, 1951. With an English translation by the editor.