Collision and intersection points

Say we have two different curves. It may be important to consider

• points the two curves share - where they intersect
• intersections which occur for the same value of ${\displaystyle t}$ - where they collide.

Intersection vector functions

We can also use vector functions to represent the curve of intersection of two surfaces. For example, we want to know the curve of intersection of the cylinder ${\displaystyle x^{2}+y^{2}=1}$ and the plane ${\displaystyle y+z=2}$.

Vector functions rely on parameterizations, so we can rewrite the equation of the cylinder into: ${\displaystyle \cos ^{2}t+\sin ^{2}t=1}$, where ${\displaystyle x=\cos t,y=\sin t}$.

From the equation of the plane, we know that ${\displaystyle z=2-y=2-\sin t}$. Thus the corresponding vector equation is:

${\displaystyle \mathbf {r} (t)=\langle \cos t,\sin t,2-\sin t\rangle }$

Limits and Continuity

The limit of a vector function ${\displaystyle \mathbf {r} }$ is defined by taking the limits of its component functions.

And the requirement for continuity is also simple:

A vector function ${\displaystyle \mathbf {r} }$ is continuous at ${\displaystyle a}$ if ${\displaystyle \lim _{t\rightarrow a}\mathbf {r} (t)=\mathbf {r} (a)}$.

Notation

There are a lot of complicated concepts surrounding vector functions. So, in order to make us easier to understand, there are several notations we need to be familiar with. Let us imagine that a vector function ${\displaystyle \mathbf {r} (t)=\langle f(t),g(t),h(t)\rangle }$. We will think it as a displacement function of a particle. For example, ${\displaystyle \mathbf {r} (t)=\langle f(t),g(t),h(t)\rangle }$ means "the location of a particle with respect to the time." If we consider the vector function as a function describing the position of a particle, we can easily understand the derivatives of the vector function:

The first derivative of the vector function: ${\displaystyle {\frac {d\mathbf {r} (t)}{dt}}=\langle f'(t),g'(t),h'(t)\rangle }$ will be notated as ${\displaystyle \mathbf {v} (t)=\mathbf {r} '(t)}$ as ${\displaystyle \mathbf {v} }$ stands for velocity. Similarly, the second derivative of the vector function will be notated as ${\displaystyle \mathbf {a} (t)=\mathbf {r} ''(t)}$ as ${\displaystyle \mathbf {a} }$ stands for acceleration.

This kind of perspective can give as an interesting insight on what is about to come in the next several sections. However, note that this notation exists only to make our understanding towards vector functions clearer and easier. It is not universal and will be used in this chapter only for the sake of convenience.

Differentiation

Recall that the first derivative of a scalar function ${\displaystyle f(x)}$ is defined as:

${\displaystyle f'(x)=\lim _{\Delta x\rightarrow 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$

The first derivative of a vector function ${\displaystyle \mathbf {r} (t)}$ is defined in much the same way:

${\displaystyle \mathbf {r} '(t)=\lim _{\Delta t\rightarrow 0}{\frac {\mathbf {r} (t+\Delta t)-\mathbf {r} (t)}{\Delta t}}}$

We can use this definition to prove that the derivative of a vector function can be presented as the derivative of its component functions.

${\displaystyle {\begin{aligned}\mathbf {r} '(t)&=\lim _{\Delta t\rightarrow 0}{\frac {1}{\Delta t}}[\mathbf {r} (t+\Delta t)-\mathbf {r} (t)]\quad {\text{definition}}\\&=\lim _{\Delta t\rightarrow 0}{\frac {1}{\Delta t}}[\langle f(t+\Delta t),g(t+\Delta t),h(t+\Delta t)\rangle -\langle f(t),g(t),h(t)\rangle ]\quad {\text{component form}}\\&=\lim _{\Delta t\rightarrow 0}\langle {\frac {f(t+\Delta t)-f(t)}{\Delta t}},{\frac {g(t+\Delta t)-g(t)}{\Delta t}},{\frac {h(t+\Delta t)-h(t)}{\Delta t}}\rangle \quad {\text{vector addition and multiplication}}\\&=\langle \lim _{\Delta t\rightarrow 0}{\frac {f(t+\Delta t)-f(t)}{\Delta t}},\lim _{\Delta t\rightarrow 0}{\frac {g(t+\Delta t)-g(t)}{\Delta t}},\lim _{\Delta t\rightarrow 0}{\frac {h(t+\Delta t)-h(t)}{\Delta t}}\rangle \quad {\text{definition}}\\&=\langle f'(t),g'(t),h'(t)\rangle \quad {\text{definition}}\\\end{aligned}}}$

Thus, using the same method, we can derive the second derivative and so on.

So the ${\displaystyle n}$th-order derivative should look like this:

${\displaystyle \mathbf {r} ^{(n)}(t)=\langle f^{(n)}(t),g^{(n)}(t),h^{(n)}(t)\rangle }$

Differentiation rules

Just like real-valued functions, there are some differentiation rules in the world of vector functions. The factor that makes vector differentiation rules slight more complicated is the product rule because there are two kinds of multiplication in vectors: dot product and cross product.

Naturally, we will prove that those rules are correct. Let us assume that ${\displaystyle \mathbf {u} (t)=\langle f_{1}(t),f_{2}(t),f_{3}(t)\rangle }$ and ${\displaystyle \mathbf {v} (t)=\langle g_{1}(t),g_{2}(t),g_{3}(t)\rangle }$.

Rule 1: the addition rule

${\displaystyle {\begin{aligned}{\frac {d}{dt}}[\mathbf {u} (t)+\mathbf {v} (t)]&={\frac {d}{dt}}[\langle f_{1}(t),f_{2}(t),f_{3}(t)\rangle +\langle g_{1}(t),g_{2}(t),g_{3}(t)\rangle ]\quad {\text{component form}}\\&={\frac {d}{dt}}\langle f_{1}(t)+g_{1}(t),f_{2}(t)+g_{2}(t),f_{3}(t)+g_{3}(t)\rangle \quad {\text{vector addition}}\\&=\langle {\frac {d}{dt}}(f_{1}(t)+g_{1}(t)),{\frac {d}{dt}}(f_{2}(t)+g_{2}(t)),{\frac {d}{dt}}(f_{3}(t)+g_{3}(t))\rangle \quad {\text{distribution}}\\&=\langle f_{1}'(t)+g_{1}'(t),f_{2}'(t)+g_{2}'(t),f_{3}'(t)+g_{3}'(t)\rangle \quad {\text{addition rule for real-valued functions}}\\&=\langle f_{1}'(t),f_{2}'(t),f_{3}'(t)\rangle +\langle g_{1}'(t),g_{2}'(t),g_{3}'(t)\rangle \quad {\text{vector addition}}\\&=\mathbf {u} '(t)+\mathbf {v} '(t)\quad {\text{vector form}}\\\end{aligned}}}$

Rule 2: the scalar multiplication rule

${\displaystyle {\begin{aligned}{\frac {d}{dt}}[f(t)\mathbf {u} (t)]&={\frac {d}{dt}}[f(t){\begin{pmatrix}f_{1}(t)\\f_{2}(t)\\f_{3}(t)\end{pmatrix}}]\quad {\text{component form}}\\&={\frac {d}{dt}}{\begin{pmatrix}f(t)f_{1}(t)\\f(t)f_{2}(t)\\f(t)f_{3}(t)\end{pmatrix}}\quad {\text{scalar multiplication}}\\&={\begin{pmatrix}{\frac {d}{dt}}(f(t)f_{1}(t))\\{\frac {d}{dt}}(f(t)f_{2}(t))\\{\frac {d}{dt}}(f(t)f_{3}(t))\\\end{pmatrix}}\quad {\text{distribution}}\\&={\begin{pmatrix}f'(t)f_{1}(t)+f(t)f_{1}'(t)\\f'(t)f_{2}(t)+f(t)f_{2}'(t)\\f'(t)f_{3}(t)+f(t)f_{3}'(t)\\\end{pmatrix}}\quad {\text{multiplication rule for real-valued functions}}\\&={\begin{pmatrix}f'(t)f_{1}(t)\\f'(t)f_{2}(t)\\f'(t)f_{3}(t)\end{pmatrix}}+{\begin{pmatrix}f(t)f_{1}'(t)\\f(t)f_{2}'(t)\\f(t)f_{3}'(t)\end{pmatrix}}\quad {\text{vector addition}}\\&=f'(t){\begin{pmatrix}f_{1}(t)\\f_{2}(t)\\f_{3}(t)\end{pmatrix}}+f(t){\begin{pmatrix}f_{1}'(t)\\f_{2}'(t)\\f_{3}'(t)\end{pmatrix}}\quad {\text{scalar multiplication}}\\&=f'(t)\mathbf {u} (t)+f(t)\mathbf {u} '(t)\quad {\text{vector form}}\\\end{aligned}}}$

Rule 3: the dot product rule

${\displaystyle {\begin{aligned}{\frac {d}{dt}}[\mathbf {u} (t)\cdot \mathbf {v} (t)]&={\frac {d}{dt}}[\langle f_{1}(t),f_{2}(t),f_{3}(t)\rangle \cdot \langle g_{1}(t),g_{2}(t),g_{3}(t)\rangle ]\quad {\text{component form}}\\&={\frac {d}{dt}}(f_{1}(t)g_{1}(t)+f_{2}(t)g_{2}(t)+f_{3}(t)g_{3}(t))\quad {\text{dot product}}\\&={\frac {d}{dt}}\sum _{i=1}^{3}f_{i}(t)g_{i}(t)\quad {\text{simplification}}\\&=\sum _{i=1}^{3}{\frac {d}{dt}}f_{i}(t)g_{i}(t)\quad {\text{distribution}}\\&=\sum _{i=1}^{3}[f_{i}'(t)g_{i}(t)+f_{i}(t)g_{i}'(t)]\quad {\text{multiplication rule for real-valued functions}}\\&=\sum _{i=1}^{3}f_{i}(t)g_{i}(t)+\sum _{i=1}^{3}f_{i}(t)g_{i}'(t)\quad {\text{vector addition}}\\&=\mathbf {u} '(t)\cdot \mathbf {v} (t)+\mathbf {u} (t)\cdot \mathbf {v} '(t)\quad {\text{vector form}}\\\end{aligned}}}$

Rule 4: the cross product rule

${\displaystyle {\begin{aligned}{\frac {d}{dt}}[\mathbf {u} (t)\times \mathbf {v} (t)]&={\frac {d}{dt}}[\langle f_{1}(t),f_{2}(t),f_{3}(t)\rangle \times \langle g_{1}(t),g_{2}(t),g_{3}(t)\rangle ]\quad {\text{component form}}\\&={\frac {d}{dt}}{\begin{vmatrix}\mathbf {\hat {i}} &\mathbf {\hat {j}} &\mathbf {\hat {k}} \\f_{1}(t)&f_{2}(t)&f_{3}(t)\\g_{1}(t)&g_{2}(t)&g_{3}(t)\\\end{vmatrix}}\quad {\text{cross product}}\\&={\frac {d}{dt}}{\begin{pmatrix}f_{2}(t)g_{3}(t)-f_{3}(t)g_{2}(t)\\f_{3}(t)g_{1}(t)-f_{1}(t)g_{3}(t)\\f_{1}(t)g_{2}(t)-f_{2}(t)g_{1}(t)\\\end{pmatrix}}\quad \\&={\begin{pmatrix}{\frac {d}{dt}}(f_{2}(t)g_{3}(t)-f_{3}(t)g_{2}(t))\\{\frac {d}{dt}}(f_{3}(t)g_{1}(t)-f_{1}(t)g_{3}(t))\\{\frac {d}{dt}}(f_{1}(t)g_{2}(t)-f_{2}(t)g_{1}(t))\\\end{pmatrix}}\quad {\text{distribution}}\\&={\begin{pmatrix}f_{2}'(t)g_{3}(t)+f_{2}(t)g_{3}'(t)-f_{3}'(t)g_{2}(t)-f_{3}(t)g_{2}'(t)\\f_{3}'(t)g_{1}(t)+f_{3}(t)g_{1}'(t)-f_{1}'(t)g_{3}(t)-f_{1}(t)g_{3}'(t)\\f_{2}'(t)g_{3}(t)+f_{2}(t)g_{3}'(t)-f_{3}'(t)g_{2}(t)-f_{3}(t)g_{2}'(t)\\\end{pmatrix}}\quad {\text{multiplication rule for real-valued functions}}\\&={\begin{pmatrix}f_{2}'(t)g_{3}(t)-f_{3}'(t)g_{2}(t)+f_{2}(t)g_{3}'(t)-f_{3}(t)g_{2}'(t)\\f_{3}'(t)g_{1}(t)-f_{1}'(t)g_{3}(t)+f_{3}(t)g_{1}'(t)-f_{1}(t)g_{3}'(t)\\f_{2}'(t)g_{3}(t)-f_{3}'(t)g_{2}(t)+f_{2}(t)g_{3}'(t)-f_{3}(t)g_{2}'(t)\\\end{pmatrix}}\quad {\text{rearrangement}}\\&={\begin{pmatrix}f_{2}'(t)g_{3}(t)-f_{3}'(t)g_{2}(t)\\f_{3}'(t)g_{1}(t)-f_{1}'(t)g_{3}(t)\\f_{2}'(t)g_{3}(t)-f_{3}'(t)g_{2}(t)\\\end{pmatrix}}+{\begin{pmatrix}f_{2}(t)g_{3}'(t)-f_{3}(t)g_{2}'(t)\\f_{3}(t)g_{1}'(t)-f_{1}(t)g_{3}'(t)\\f_{2}(t)g_{3}'(t)-f_{3}(t)g_{2}'(t)\\\end{pmatrix}}\quad {\text{vector addtion}}\\&={\begin{vmatrix}\mathbf {\hat {i}} &\mathbf {\hat {j}} &\mathbf {\hat {k}} \\f_{1}'(t)&f_{2}'(t)&f_{3}'(t)\\g_{1}(t)&g_{2}(t)&g_{3}(t)\\\end{vmatrix}}+{\begin{vmatrix}\mathbf {\hat {i}} &\mathbf {\hat {j}} &\mathbf {\hat {k}} \\f_{1}(t)&f_{2}(t)&f_{3}(t)\\g_{1}'(t)&g_{2}'(t)&g_{3}'(t)\\\end{vmatrix}}\quad {\text{cross product}}\\&={\begin{pmatrix}f_{1}'(t)\\f_{2}'(t)\\f_{3}'(t)\\\end{pmatrix}}\times {\begin{pmatrix}g_{1}(t)\\g_{2}(t)\\g_{3}(t)\\\end{pmatrix}}+{\begin{pmatrix}f_{1}(t)\\f_{2}(t)\\f_{3}(t)\\\end{pmatrix}}\times {\begin{pmatrix}g_{1}'(t)\\g_{2}'(t)\\g_{3}'(t)\\\end{pmatrix}}\quad \\&=\mathbf {u} '(t)\times \mathbf {v} (t)+\mathbf {u} (t)\times \mathbf {v} '(t)\quad {\text{vector form}}\\\end{aligned}}}$

Rule 5: the chain rule

${\displaystyle {\begin{aligned}{\frac {d}{dt}}[\mathbf {u} (f(t))]&={\frac {d}{dt}}{\begin{pmatrix}f_{1}(f(t))\\f_{2}(f(t))\\f_{3}(f(t))\\\end{pmatrix}}\quad {\text{component form}}\\&={\begin{pmatrix}{\frac {d}{dt}}f_{1}(f(t))\\{\frac {d}{dt}}f_{2}(f(t))\\{\frac {d}{dt}}f_{3}(f(t))\\\end{pmatrix}}\quad {\text{distribution}}\\&={\begin{pmatrix}f'(t)f_{1}'(f(t))\\f'(t)f_{2}'(f(t))\\f'(t)f_{3}'(f(t))\\\end{pmatrix}}\quad {\text{chain rule for real-valued functions}}\\&=f'(t){\begin{pmatrix}f_{1}'(f(t))\\f_{2}'(f(t))\\f_{3}'(f(t))\\\end{pmatrix}}\quad {\text{scalar multiplication}}\\&=f'(t)\mathbf {u} '(f(t))\quad {\text{vector form}}\\\end{aligned}}}$

Continuity & differentiability

If we have a parametrization ${\displaystyle {\vec {f}}:\mathbb {R} \to \mathbb {R} ^{n}}$ , which is built up out of component functions in the form ${\displaystyle {\vec {f}}(t)={\big (}f_{1}(t),\ldots ,f_{n}(t){\big )}}$ , ${\displaystyle {\vec {f}}}$ is continuous if and only if each component function is also.

In this case the derivative of ${\displaystyle {\vec {f}}(t)}$ is

${\displaystyle a_{i}={\big (}f_{1}'(t),\ldots ,f_{n}'(t){\big )}}$ . This is actually a specific consequence of a more general fact we will see later.

Tangent vectors

Recall in single-variable calculus that on a curve, at a certain point, we can draw a line that is tangent to that curve at exactly at that point. This line is called a tangent. In the several variable case, we can do something similar.

We can expect the tangent vector to depend on ${\displaystyle {\vec {f}}'(t)}$ and we know that a line is its own tangent, so looking at a parametrised line will show us precisely how to define the tangent vector for a curve.

An arbitrary line is ${\displaystyle {\vec {f}}(t)=t{\vec {a}}+{\vec {b}}}$ , with: ${\displaystyle f_{i}(t)=ta_{i}t+b_{i}}$ , so

${\displaystyle f_{i}'(t)=a_{i}}$ and

${\displaystyle f'(t)={\vec {a}}}$ , which is the direction of the line, its tangent vector.

Similarly, for any curve, the tangent vector is ${\displaystyle f'(t)}$ .

Angle between curves

We can then formulate the concept of the angle between two curves by considering the angle between the two tangent vectors. If two curves, parametrized by ${\displaystyle {\vec {f_{1}}}}$ and ${\displaystyle {\vec {f_{2}}}}$ intersect at some point, which means that

${\displaystyle {\vec {f_{1}}}(s)={\vec {f_{2}}}(t)=c}$

the angle between these two curves at ${\displaystyle c}$ is the angle between the tangent vectors ${\displaystyle {\vec {f_{1}'}}(s)}$ and ${\displaystyle {\vec {f_{2}'}}(t)}$ is given by

${\displaystyle \arccos \left({\frac {{\vec {f_{1}'}}(s)\cdot {\vec {f_{2}'}}(t)}{{\Big \|}{\vec {f_{1}'}}(s){\Big \|}{\Big \|}{\vec {f_{2}'}}(t){\Big \|}}}\right)}$

Different parameterizations

One such parametrization of a curve is not necessarily unique. Curves can have several different parametrizations. For example, we already saw that the unit circle can be parametrized by g(t) = (cos at, sin at) such that t ∈ [0, 2π/a).

Generally, if f is one parametrization of a curve, and g is another, with

f(t₀) = g(s₀)

there is a function u(t) such that u(t₀)=s₀, and g'(u(t)) = f(t) near t₀.

This means, in a sense, the function u(t) "speeds up" the curve, but keeps the curve's shape.

Surfaces

A surface in space can be described by the image of a function f : R² → Rⁿ. f is said to be the parametrization of that surface.

For example, consider the function

f(α, β) = α(2,1,3)+β(-1,2,0)

This describes an infinite plane in R³. If we restrict α and β to some domain, we get a parallelogram-shaped surface in R³.

Surfaces can also be described explicitly, as the graph of a function z = f(x, y) which has a standard parametrization as f(x,y)=(x, y, f(x,y)), or implictly, in the form f(x, y, z)=c.