Geometric Transformations of the PlaneLecture 6 Feb 14, 2021
Description of the idea
o Idea. We will study basic moves that change shapes in plane
o We learn how these moves change basic shapes such as a line or circle, or quantities such as length and angle
o Finally, we combine these moves to make more complicated moves
We call these moves geometric transformations
Examples
(1) Translation: We move every point in certain direction, with certain distance.
- A translation π»π is described by a vector π: π β π»π(π) = π + π
- Translations preserve length and angle
- It does not fixes any point unless π = 0
- If π = 0, then T does not make any change
π
π»π(π)
π
Doing multiple moves
- Description: If you first move points with a vector π’ and then with a vector π£,
it is as if you have moved points with sum of two vectors π’ + π£
- Mathematical notation: π" β π# = π#$"
WE SAY : Composition of a series of
translations by vectors π£%, β¦ , π£& is a
translation by the vector π£ = π£% +β―+ π£&
π
π»π π»π π = π»π#π (π)π
π
π»π(π)
π + π
Examples
(2) Homothecy (or Homothety): A transformation of plane which dilates distances with respect to a fixed point (center of Homothecy)
- A Homothecy π―π,π is described by its center point
π and the dilation factor π:
π β π―π,π(π) = π + π (π β π)
- Homothecy preserve angle but
changes length by a factor of r
- It only fixes the center point π (unless π = 1 )
- If π = 1, then π» does not make any change
π―π,π(π)
π
π
π
π―π,π(π)
More pictures
Β§ π > 1 : things get bigger. π < 1 βΆ things get smaller
Β§ Effect of Homothecy on
Lines:
(1) every line L will be mapped to another line parallel to L
(2) If L passes through the center it will be preserved
Circles:
every Circle C will be mapped to another circle
π
Composition of two Homothecies
Β§ If we first do a Homothecy π»% = π»*',+' with center π% and dilation factor π%, and then do another a Homothecy π», = π»*(,+( with center π, and dilation factor π,, what is the over all transformation?
Β§ Mathematical notation: What can we say about π» = π», β π»% ?
Β§ Observations:
Β§ (1) π» does not change angles.
Β§ (2) π» changes distances by a total factor of π = π% π,(Q) Could H possibly be a Homothecy as well with a dilation factor of π
(Q) If yes, where will be its center?
Where is the center of π»?
Β§ Lemma: If π%π, = 1, then the composition of π»% and π», is a translation
by the vector π£ = π, π%π,Β§ Lemma: If π%π, β 1, then the composition of π»% and π», is a homothety with ratio π = π%π,
and the center π will be somewhere on the line π%π,Β§ Four cases:
(i) π, π, > 1, (ii) π, π, < 1, (iii) π < 1 < π,, (iV) π, < 1 < π
ππ
ππ
Lets rework a problem form last time
Strategy of the proof
Β§ 1. We perform 3 homotheties with centers π₯, π¦ , π§ and ratios +*++, +,+*, πππ ++
+,respectively.
Β§ 2. The circle C is fixed in this process
Β§ 3. The overall ration is 1.
Β§ 4. So the composition of these 3 homotheties does not move any point
Now
Β§ 5. Track the line passing through π₯π¦ in this process to conclude that π§ must be on it as well.
Proof of Menelaus (version 2) with homothety
Theorem. Suppose --.
--/Γ .-/.--
Γ /--/-.
= 1 then
Aβ, Bβ, Cβ are on one line L (we say Aβ, Bβ, Cβ are collinear)
Repeat the same proof as the previous one.
A
B CAβ
BβCβ
Problems to solve with Homothecy
Β§ https://www.cut-the-knot.org/pythagoras/Transforms/ProblemsByHomothety.shtml
Β§ https://eldorado.tu-dortmund.de/bitstream/2003/31745/1/195.pdf
Β§ http://users.math.uoc.gr/~pamfilos/eGallery/problems/Similarities.pdf
Homothety with negative scaling factor
We can also consider Homotheties π―(π,π) where the dilation factor π < 0:
π β π―π,π(π) = π + π (π β π)
The same rule as before applies:
Composition of Homotheties is a homothety
(sometimes a translation) π―π,.π(π)
π
π
ππ―π,.π(π)
Proof of Menelaus (version 1) with Homothety
Theorem. Suppose --.
--/Γ .-/.--
Γ /--/-.
= 1 then
Aβ, Bβ, Cβ are on one line L (we say Aβ, Bβ, Cβ are collinear)A
B CAβ
BβCβ
More geometric transformations
Β§ (3) Reflection across a line L:
Β§ Properties:
- Applying π 2 twice, we get identity
- It preserves distance and angle
Lπ
πΉπ³(π)
Composition of two reflections
Β§ (3) If we reflect across a line L and then Lβ
what is the composition of two reflections?L
π
π = πΉπ³(π)
Lβπ = πΉπ³0(π)
More geometric transformations
Β§ (4) Rotation around a point π with angle π
Β§ Properties:
- It preserves distance and angle
- The only point fixed is the center π
π
πΉπ,π½(π)
π
π½
More geometric transformations
Β§ (5) Inversion with respect to a point π and product distance π, :
It takes any point π₯ besides the center π to another point π¦ = πΌ π₯
Colinear with π₯ and π such that ππ₯ Γ ππ¦ = π,
It changes any line not passing through the center π to
a circle (as in the picture). In this regard, it is different from
transformations we had before.
π =
π
π
π°(π)
π
π