# Transformations: Scaling, Moving, Rotating a Curve

## Vectors

A point {a,b} has a distance to the origin. Also, it makes a angle with the positive x-axis if we consider the line {0,0} to {a,b}. These two concepts are very useful and convenient, that we define the length of a point to be the distance from the origin to the point, and we define the angle of a point to be the angle it makes with the positive x-axis. When a point is considered this way, we call it a **vector**. We say, that a vector has a length, and a angle.

If a point A has coordinates {a,b}, its length can be expressed as `Sqrt[a^2+b^2]`

. This comes easily from
Pythagorean theorem.

The angle of a vector {a,b} is `ArcTan[b/a]`

. This is from trigonometry.

## Scaling a Vector

Given a vector {a,b}, to scale (dilate/contract) the vector by s around the origin, the new vector is `{a,b}*s`

.

### Scaling a Curve

Let
`f[x,y]==0`

be the equation for a curve in rectangular coordinates.
To scale the curve by `s`

, the new equation would be:
`f[x/s, y/s]==0`

Let
`f[θ,r]==0`

be the equation for a curve in polar coordinate.
To scale it by `s`

, the new formula is
`f[r/s,θ]==0`

.

Let
`{f[t],g[t]}`

be the parametric formula for a curve in rectangular coordinate.
To scale it by `s`

, the new formula is
`{f[t],g[t]}*s`

.

Let
`{r[t],φ[θ]}`

be the parametric formula for a curve in polar coordinate.
To scale it by `s`

, the new formula is
`{r[t]*s,φ[θ]}`

.

#### Example

A circle `(x-1)^2+y^2-1==0`

has center `{1,0}`

and radius `1`

. To scale it by a factor of `1/2`

, we replace `x`

by `2*x`

and `y`

by `2*y`

, to obtain
`(2*x-1)^2+(2*y)^2-1==0`

.

## Translate a Vector

Given a vector {a,b}, to translate the vector by {c,d}, the new vector is {a+c,b+d}.

The adding of coordinate components is a very convenient operation. We define it as vector addition. That is, `{a,b}+{c,d} := {a+c,b+d}`

, and its geometric interpretation is that one vector is moved to a new position by the second vector.

### Translate a Curve

Let
`f[x,y]==0`

be the equation for a curve in rectangular coordinates.
To translate the curve by vector {a,b}, the new equation would be:
`f[x-a, y-b]==0`

.

Let
`{f[t],g[t]}`

be the parametric formula for a curve in rectangular coordinate.
To translate it by vector {a,b}, the new formula is
`{f[t],g[t]} + {a,b}`

## Rotate a Vector

The point `{Cos[α],Sin[α]}`

is a point with α angle with the positive x-axis and 1 unit distant from the origin. To rotate it by angle β, then new coordinate is `{Cos[α+β],Sin[α+β]}`

. By trig identity of double angle:

Cos[a+b] == Cos[a] Cos[b] - Sin[a] Sin[b]

and

Sin[a+b] = Cos[a] Sin[b] + Cos[b] Sin[a]

, we can then write our rotated point as

{Cos[α] Cos[β] - Sin[α] Sin[β], Cos[α] Sin[β] + Cos[β] Sin[α]}

Now, look at the components in the above coordinate. They are of the form Sin[…] or Cos[…]. In other words, if we have a point `A := {Cos[α],Sin[α]}`

and point `B := {Cos[β],Sin[β]}`

.
Then, the point A rotated by β angles can be written in terms of the coordinate components of A and B.

In other words, if A is {a,b} and B is {c,d} and suppose both are 1 unit distant from the origin, then a new point C, obtained by rotating A by B's angle, can be expressed in terms of coordinate components of A and B by this expression: `{a c-b d, a d+b c}`

.

But now if we substitute `a`

by `r*a`

and `b`

by `r*b`

and `c`

by `s*c`

and `d`

by `s*d`

. In other words, we start with points `{r a,r b}`

which is `r`

distant to the origin, and point `{s c,s d}`

which is `s`

distant. Then, the above formula gives us `{r s (a c-b d), r s (a d+b c)}`

. From this, we can say that length of the new vector is just the product of the lengths of the old vectors.

In summary, if A is {a,b} with angle α and length r, and B is {c,d} with angle β and length s, then `{a c-b d, a d+b c}`

is a point with angle `α+β`

and length `r*s`

.

This formula is extremely powerful, because it lets us do rotation and scaling around the origin at the same time, and by simply using another vector.

#### Note About Complex Numbers

Note: rotating one vector by another, is exactly how multiplication of 2 complex numbers is defined. You can think of complex number `a+b ⅈ`

as vector `{a,b}`

. The definition of multiplication of 2 complex numbers is exactly the definition of rotating one vector by another. That is, the multplication of complex number `a+b ⅈ`

and `c+d ⅈ`

is defined to be `{a c - b d, (a d + b c) ⅈ}`

. You can see that it's the same as rotation of vectors, expressed by the vector's coordinate components. 〔►see Understanding Complex Numbers〕

### Rotating a Curve defined by a Equation

Let `f[x,y]==0`

be the equation for a curve in rectangular coordinates.

Suppose we want to rotate it by a angle θ. First, we find a vector {c,d} of distance 1 having angle -θ, which is `{Cos[-θ], Sin[-θ]}`

. Then, we do this substitution into the function:

x → x c - y d y → x d + y c

so we obtain:

f[x c - y d, x d + y c]

This would be the new curve we wanted. If the vector {c,d} has length r, then the new curve would be dilated by `1/r`

.

#### Example

Suppose:

f[x,y] := (x-1)^2+y^2-1

Then, `f[x,y]==0`

is a circle centered on {1,0} with radius 1.

Let's say we want to rotate it by angle θ represented by the vector {2,1}.
So, we should use a vector that has angle -θ, which is {2,-1}.
So, `{c,d}:={2,-1}`

.
Now, we substitute the rotation with our formula

x → x 2 - y (-1) y → x (-1) + y 2

So our new function is:

g[x,y] := ((x 2 - y (-1))-1)^2 + ( x (-1) + y 2)^2-1

Then, `g[x,y]==0`

is our old circle rotated. Note that it is also shrinked. This is because the vector we used {2,-1} has a length greater than 1. If we want to keep the circle the same size, we should use a vector with length 1, which is `{Cos[-θ],Sin[-θ]}`

.

Let
`f[r,θ]==0`

be the equation for a curve in polar coordinate.
To rotate it by α, the new formula is
`f[r,θ-α]==0`

.

Let
`{f[t],g[t]}`

be the parametric formula for a curve in rectangular coordinate.
To rotate the curve by θ radian, the new formula is
`{Cos[θ]*f[t] - g[t]*Sin[θ], Cos[θ]*g[t] + f[t]*Sin[θ]}`

or, if {c,d} is a unit vector with θ radians,
the new rotated curve expressed in terms of {c,d} is
`{f[t]*c-g[t]*d, f[t]*d+g[t]*c}`

.

Let
`{r[t],φ[t]}`

be the parametric formula for a curve in polar coordinate.
To rotate the curve by `θ`

, the new formula would be:
`{r[t],φ[t]+θ}`

.