Dilatations - Coordinate View

A dilatation in the plane depends on a non-zero scale factor k, and is given by a mapping

[X,Y] =  f_k(x,y) = (kx, ky)

When k is larger than 1 in magnitude, that is, when |k| > 1, the dilatation, is an expansion. It moves points further away from the origin and from each other. Exercise: show why.

Remark: In Quebec textbooks, the dilatation scale factor k is allowed to be negative. 

Example:  For k =2, 

f_k(x,y) = [2x, 2y]

f_k(3,4) = [2*3, 2*4] = [6,8]

and f_k(-3,1) = [2*(-3), 2*1] = [-6,2]

When k is smaller than 1 in magnitude, when 0< |k| < 1, the dilatation, the dilatation is a contraction. It moves points closer to the origin and to each other. Exercise: show why

Example:  For k =¾, 

f_k(x,y) = [(¾)x, (¾)y]

f_k(4,-8) = [(¾)*4, (¾)*(-8)] = [3,-6]

and f_k(-3,1) = [(¾)*(-3), (¾)*1] = [-9/4,¾]

When k equals 1 in magnitude, when |k| = 1, the dilatation is  reflection in the case k = -1 or the identify map in the case k=1. The distance between the images of a pair of points equal the distance between the points. 

Example:  For k =1, 

f_k(x,y) = [(1)x, (1)y] = [x,y] 

f_k(4,-8) = [4,-8]

and f_k(-3,1) = [(1)*(-3), (1)*1] = [-9/4,-1]

Example:  For k =-1, 

f_k(x,y) = [(-1)x, (-1)y]

f_k(4,-8) = [(-1)*4, (-1)*(-8)] = [-4,8]

and f_k(-3,1) = [(-1)*(-3), (-1)*1] = [3,-1]

Distances and Lengths are Multiplied by |k|

The distance c between the points [x₁,y₁] and [x₂,y₂] 

is given by

Therefore the distance between the images of these point under a dilatation 

f_k(x,y) = (kx, ky)

that is the distance d between  the points [X₁,Y₁] =[kx₁,ky₁] and [X₂,Y₂] = [kx₂,ky₂] is given by

In the discussion of square roots, we learn that root of a product of non-negative terms is a product of the square root of the terms. 

Lines go into Lines

Theorem:  If [x,y] satisfying ax+by=c then [X,Y}=[kx,ky] satisfy the equation aX+bY = c/k

Proof:  c = ax+by = a (X/k) + b (Y/k) = aX (1/k) + bY (1/k) = (aX+bY)(1/k) since division by k gives the same result as multiplication by 1/k, and since multiplication by a real number, here 1/k is distributive. 

The above theorem say straight lines are taken into straight lines by dilatations. We observe the image line is parallel to the original line.

Theorem:  If [x,y] satisfying ax+by=c then [X,Y}=[kx,ky] satisfy the equation aX+bY = c/k

Line Segments Go Into Line Segments

The line segment between  the points [x₁,y₁] and [x₂,y₂] can be described algebraically by the parameter-based equations

[x,y] = (1-t) [x₁,y₁] + t [x₂,y₂] 
          = [(1-t) x₁  + t x₂, (1-t) y₁+ ty₂]

where 0 < t < 1, that is t varies between 0 and 1.  

The value t = 0 gives the initial end  [x₁,y₁] while the  value t = 1 gives the other end [x₂,y₂], and the value t = ½ gives the midpoint. [(x₁+ x₂)/2, (y₁+y₂)/2 ]. 

If we let the points [X₁,Y₁] =[kx₁,ky₁] and [X₂,Y₂] = [kx₂,ky₂] be the image of the two end points  [x₁,y₁] and [x₂,y₂] with respect to a dilatation of scale factor k then  the image of 

[x,y] = [(1-t) x₁  + t x₂, (1-t) y₁+ ty₂]    
         = (1-t) [x₁,y₁] + t [x₂,y₂] 

is 

[X,Y] = [(1-t) X₁  + t X₂, (1-t) Y₁+ tY₂]
           = (1-t) [X₁,Y₁] + t [X₂,Y₂] 

That is because  

[X,Y]

= [kx,ky] 
= [k{(1-t) x₁  + t x₂}, k{(1-t) y₁+ ty₂}]
= [  (1-t) kx₁  + t kx₂, (1-t)k y₁+ tky₂}]
=  [(1-t) X₁  + t X₂, (1-t) Y₁+ tY₂]

So the image of  

[x,y] =  (1-t) [x₁,y₁] + t [x₂,y₂]

 is 

[X,Y] = (1-t) [X₁,Y₁] + t [X₂,Y₂] 

From above the length of the image line segment between point  [X₁,Y₁] =[kx₁,ky₁] to point  [X₂,Y₂] = [kx₂,ky₂] is absolute value of the scale factor |k| times the distance of the original line segment between points  [x₁,y₁] and [x₂,y₂]. So a dilatation with scale factor k mutliples lengths by |k|

The image of a line segments is a parallel line segments with length |k| times the original.  

Preservation of Angles

An angle is formed when two line segments meeting at a common endpoint. The image of those line segments are parallel line segments. The foregoing suggests the angle of image line segments equals the angle of formed by the original.  

Similar Polygons and Similar Triangles

Two polygons in the plane are said to be similar when and only when corresponding angles are equal and corresponding sides are proportional.  To be more precise, there is a scale factor K for which length of a side in the second polygon are K times the length of the corresponding side in the first.

Now the image of a first polygon under a dilatation scale factor k is a second polygon with the following properties:

• Each  side in the second has length k times the corresponding side in the first. The correspondence is provided by the dilatation.
• The angle formed by two sides in the second equals the angle formed by the corresponding two adjacent sides in the first. 

So the image of a first polygon under a dilatation is a similar polygon,  a similar polygon with corresponding sides parallel.

Maps, Plans and Similarity

A dilatation [X,Y] =  f_k(x,y) = (kx, ky) represents  a contraction, expansion or identify map in a plane if [X,Y} and [x,y] determine points in the same coordinate system. But a A dilitation [X,Y] =  f_k(x,y) = (kx, ky) may also represent a function between two different planes, the original and the image. The image may be regarded as a map (the kind used for roads, cities and geographic regions in a flat earth approximation society). 

Many human constructions and roads are based on polygons or can approximated by polygons. In a map, corresponding figures are similar and perhaps parallel. Corresponding similar figures can be made parallel if you rotate the map. Producing a map of a flat region preserves angles while it multiples lengths by a scale factor k >0.  The ratio 1: 100 corresponds to a map with scale factor k = 1/100. That is lengths in the map are 1/100 lengths of the original. 

Scale Factors for lengths, areas and volumes.

Plans and models, if not maps, can be 3 dimensional. Curves, Regions and Solids (or their surfaces) can be drawn with scale factor k (a ratio). In such plans and models, angles are preserved (the angle of the image or model equals the angle in the original or intended original). While lengths are multiplied by k, areas are multiplied by a = k² and volumes are multiplied by b = k³.  The foregoing has implications for the amounts proportional to lengths, surface area and volume. If you know any one of the three scale factors k, a and b, the other two can be computed. 

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