Dot Product

The dot product between two vectors is an important quality given by (a,b,c)⋅(d,e,f) = ad + be + cf. This has a few important properties:
1) α⋅α=|α|2
2) If we consider the zero vector to be orthogonal to all vectors, then α⋅β = 0 if and only if α is orthogonal to β
3) α⋅β=|α||β|cosθ where θ the angle between the two vectors.
The dot product is an example of an inner product, which has importance for vectors in all sorts of vector spaces.





Cross Product

The cross product [x] of two vectors (a, b, c) and (d, e, f) is the vector (bf - ce, cd - af, ae - bd) .

The magnitude of the cross product equivalent to the product of the magnitudes of the two vectors times the sine of the angle between them (starting from the first). The direction is determined by the right-hand rule: if you give a "thumbs-up", and your fingers curl in the direction that the first vector needs to rotate to get to the second vector, then your thumb will point in the direction of the cross product.

The magnitude of the cross product of (a, b, c) and (d, e, f) is equivalent to the area of the parallelogram determined by the vectors (a, b, c) and (d, e, f).

Note that the cross product is not commutative. In particular, A x B = -B x A.




Triple Vector Product

The triple vector product of three vectors A = (a, b, c), B = (d, e, f), and C = (g, h, i) is the scalar (A x B)⋅C = (bf - ce)g + (cd - af)h + (ae - bd)i.

The triple vector product gives the area of the parallelopiped determined by the three given vectors.  The volume of a parallelpipedal solid is the height multiplied by the area of the base, and the height is the length of the projection of C to the line along A x B.  Note that the quantity (A x B)⋅C can be either positive or negative, so it is actually a "signed volume" which is positive if C lies above the oriented plane determined by A and B in that order, and negative if it lies below that oriented plane.




Parametrized Space Curves

A parametrized curve X(t) in space is a collection of points X(t) = (x(t),y(t),z(t)) where the coordinate functions x(t), y(t), and z(t) are functions defined over a domain in the real line, usually an interval of the form [a,b] = {t | a ≤ t ≤ b}, where a or b can be infinite.

When the coordinate functions x(t), y(t), and z(t) are not only continuous but also differentiable, then we may form the "velocity vector" X'(t) = (x'(t),y'(t),z'(t)). If the coordinate functions are twice differentiable, we may form the acceleration vector X''(t) = (x''(t),y''(t),z''(t)).

Note that parameterized plane curves are special cases of parameterized space curves.




Arclength and Arclength Reparametrization

The arclength of a curve is given by s(t) = ∫at |X'(u)|du where |X'(t)| = √((x'(t)2 + y'(t)2)+z'(t)2). Thus we define the speed of X(t) to be s'(t) = |X'(t)|. Define L(X) to be the length of a curve.

Given a function h(t): [a,b] --> [a,b] we may reparameterize the curve X(t) by Y(t) = X(h(t)). Clearly Y(t) will be a subset of X(t). Thus Y'(t) = h'(t)X'(h(t)). In particular, one may in theory choose an arclength reparameterization of X(t) by choosing h(t) such that h'(t) = 1/|X'(h(t))| = 1/(s'(t)h(t)). Then |Y'(t)| = 1, so Y'(t) is a unit vector.  Reparametrization leaves L(X) unchanged.





Frenet Frame

We say that a curve X(t) is regular if for all t in the domain of X(t), X'(t) exists and is non-zero. If X'(t) = 0, then the curve is said to be singular at t. If a curve is regular, we may define the unit tangent vector T(t) = X'(t)/|X'(t)|. Note that X'(t) = s'(t)T(t) and |T(t)| = 1, so T(t) is in fact a unit vector.

When we wish to work in an orthonormal frame based at a point X(t) = (x(t),y(t)), the natural choice for our first vector in the frame is the unit tangent vector.

We choose a second vector, the Principal Normal Vector, by P(t) = T'(t)/|T'(t)|. Note that this is perpendicular to T(t) because T(t) was a vector of constant length, and P(t) has unit length as long as it exits.

Choosing the final vector is easily accomplished by taking the Binormal Vector, B(t) = T(t) x P(t).




Curvature and Torsion

We define the curvature of a regular curve X(t) to be κ(t) = ||T'(t)||/s'(t).  Using this definition of curvature, we see that T'(t) = s'(t)κ(t)P(t).

Recall that B(t) = T(t) x P(t), so B'(t) = T'(t) x P(t) + T(t) x P'(t) = T(t) x P'(t). Since P(t) is a unit vector, P'(t) is perpendicular to P(t), so P'(t) = f(t)T(t) + g(t)B(t) for some continuous functions f(t) and g(t). Thus B'(t) = T(t) x (f(t)T(t) + g(t)B(t)) = T(t) x g(t)B(t) = -g(t)P(t). We define torsion τ(t) by B'(t) = s'(t)τ(t)P(t). Intuitively, torsion may be thought of as the amount that the curve twists away from being planar.

Using the above definitions and some calculation left to the reader we arrive at the following relation between the Frenet frame and its derivative:


Where T(t), P(t), and B(t) are considered as column vectors.



Fundamental Theorem for Space Curves

Given curvature κ(t), torsion τ(t), and speed s'(t), up to rotation and translation these determine a unique space curve X(t). Thus, given X(0), T(0), κ(t), τ(t), and s'(t), the plane curve X(t) is uniquely determined. The proof of this fact relies on results concerning existence and uniqueness of solutions to differential equations.




Horn's Lemma

Lemma: Given a regular curve C on the unit sphere, if C has length less than 2π then C is contained in a hemisphere.

Proof:  Assume that C has length less than 2π. Begin by reparametrizing the curve by arclength, so that C is parametrized by X: [0,l] --> S2. Pick two points P and Q such that P=X(0)=X(l) and Q=X(l/2). Note that in fact any two points of maximum distance apart on the curve will work. Choose point M=(P+Q)/2. We claim that M is a north pole to a hemisphere containing C.

Assume not. Define C1 to be the curve from P to Q in the direction of C, and C2 to be the curve from Q to P in the direction of C. Without loss of generality, assume that C1 intersects the equator to our chosen hemisphere. Rotate curve C1 on the sphere about the north pole π radians to a new curve C1'. Then C1∪ C1' is a closed curve that intersects the equator at two antipodal points. Then l(C1 ∪ C1') ≥ 2π, but l(C1') = l(C1) = l(C2) so l(C) = l(C1) + l(C2) = l(C1) + l(C1') = l(C1 ∪ C1') ≥ 2π is a contradiction.




Fenchel's Theorem

Theorem: The total curvature of a regular closed space curve C is greater than or equal to 2π.

Proof:  Let X: I --> R3 be a parametrization of C, and let P be a point on the unit sphere. Consider g(t) = P⋅X(t). Because g is continuous on a compact set it attains its maximum and minimum, so for some t g'(t) = P⋅X'(t) = 0 implying that P⋅T(t) =0. Now consider that the points x ∈ S2 such that x⋅P = 0 are points lying on the equator with P as the north pole. P was chosen arbitrarily, so the tangential indicatrix intersects every great circle, and by Horn's Lemma the length of the tangential indicatrix is at least 2π. But ∫C κ(s)ds = ∫I κ(t)s'(t)dt = ∫I ||T'(t)||dt ≥ 2π.




Inscribed Quadrilateral Lemma

Lemma: Given four side lengths a,b,c,d such that it is possible to construct a quadrilateral, the quadrilateral with these side lengths and maximal area is inscribed in a circle.

Proof: 


Referring to the above picture, let A be the area of the figure. Then A=(1/2)adsin(u)+(1/2)bcsin(v).
The law of cosines gives a2+d2-2adcos(u) = e2 = b2+c2-2bccos(v).

Note that a choice of u determines v, so we can call v = v(u). Now we may consider both of the above equations as depending only on u, and differentiate both of them with respect to u resulting in
2A'(u) = adcos(u) + bcsin(v(u))v'(u) and 2adsin(u) = 2bcsin(v(u))v'(u).

To maximize A(u) we take 2A'(u) = 0 = adcos(u) + bcsin(v(u))v'(u) ⇒ adcos(u)sin(v(u)) = -bc cos(v(u))sin(v(u))v'(u) = -adsin(u)cos(v(u)), so
cos(u)sin(v(u)) + sin(u)cos(v(u)) = 0 = sin(u + v(u)), so
u+v=π which is the necessary and sufficient condition for the quadrilateral to be inscribed in a circle.


Isoperimetric Inequality

Theorem:  Given a simple closed plane curve C with length l, and A the area bounded by C, then l2 ≥ 4πA with equality if and only if C is a circle.

We Prove the following weaker version of the theorem using the inscribed quadrilateral lemma.

Theorem: Given any curve that is not a circle, it is possible to increase the ratio of area to length squared. In other words, if there is a figure with the best ratio of area to length squared then it must be a circle.

Proof:  We may start by assuming that the curve we are considering is convex, otherwise we may take the convex hull to increase area and decrease length.

Assuming that the curve C considered is not a circle, we may pick four points on C not lying on a circle. Connecting these four points we form a quadrilateral dividing the area enclosed by C into five regions, one of which is the interior of the quadrilateral. We may then move the four points and the segments of C between them forming a new curve C', preserving the side lengths of the quadrilateral as well the lengths of the segments of C. This increases the area enclosed in the quadrilateral by the inscribed quadrilateral lemma, and does not decrease the area of any of the other parts of the interior of C' while the length of C' is the same as the length of C. Thus we have increased the ratio of area to length squared.