Homothety

A homothy is a geometric transformation that can be understood as a particular case of homology, with the wrong axis and the center of the homology itself.

Definition

Be B a vector space on a body K{displaystyle scriptstyle mathbb {K} }. Be X an element (see as a point) of E. The homocy of downtown C and reason k, denoted hC,k{displaystyle scriptstyle h_{C,k}} sends a M point of space over the M' point such that:

(1a)M♫− − C=k(M− − C){displaystyle M'-C=k(M-C),}

The above can also be an affine transformation of the form:

(1b)M♫=kM+(1− − k)C{displaystyle M'=kM+(1-k)C,}

The previous relation can be written vectorially in the plane as:

[chuckles]mx♫mand♫1]=[chuckles]k0(1− − k)cx0k(1− − k)cand001][chuckles]mxmand1]{displaystyle {begin{bmatrix}m'_{x}{x}m'_{y}{bmatrix}}}={begin{bmatrix}k fake(1-k)c_{x}{x}{}{1-k)c_{y}{bmatrix}{x{x}{1b}{x{x{x}{x}{1}{b}{b

Where: M♫=(mx♫,mand♫){displaystyle M'=(m'_{x},m'_{y}),}, M=(mx,mand){displaystyle M=(m_{x},m_{y}),} and C=(cx,cand){displaystyle C=(c_{x},c_{y}),}.
In three or more dimensions the previous formula is trivially generalized.

The dilation is a composition of a linear transformation and a translation, and therefore preserves:

1. alignment: images of aligned points are aligned: (A,B,C) and (A', B', C') in the figure
2. the center of a segment, and more generally the baricentro: the image of the baricentro is the baricentro of the images. In the figure, B is the center of [A;C] and therefore B' is that of [A';C']
3. The line image is another line parallel to the original.
4. parallelism: two parallel lines have parallel images. In the figure (B'E') // (C'D') because (BE) //(CD).
   
5. If k ì 1, the center of homotecia is the only fixed point (k = 1 corresponds to the identity of E: all points are fixed).
6. k = - 1 corresponds to a symmetry of center C.
7. If it was 0, hC,k{displaystyle scriptstyle h_{C,k}} supports reciprocal transformation hC,1/k{displaystyle scriptstyle h_{C,1/k}} (when k = 0, it's not bijective).
8. When composing two homothecias of the same center you get another homotecia with this center, whose reason is the product of the reasons of the initial homothecias: hC,k{displaystyle scriptstyle h_{C,k}} or hC,k♫{displaystyle scriptstyle h_{C,k'}} = hC,k⋅ ⋅ k♫{displaystyle scriptstyle h_{C,kcdot k'}}.
9. When composing homothecias of different centers, of k and k' reasons, you get a k·k' reason homothecia when k·k'ì1, and a translation if k·k'=1. The set of homothecias (with k0) and the translations form a group.

When K is greater than zero is k greater When the body of scalars are the Reals, it is true that:

1. all lengths are multiplied by Șk, the absolute value of reason.
2. the quotient of lengths is preserved: A'C'/B'E' = AC/BE in the figure
3. oriented angles are preserved, in particular straight angles. It's obvious in the figure.

Furthermore:

1. k = - 1 corresponds to the symmetry of center C which is the rotation around C of angle π radianes (180o).
   
2. Șk 1 implies an enlargement of the figure.
   
3. أعربية Русский 1 implies a reduction.
   
4. k ≤ 0, homothecia can be expressed as the composition of a symmetry with a homothecia of LICKINE, both of the same center. That the original homotecia.

Dilations in the real plane

In this section, the scalars will be real numbers.

A generalized dilation in the plane is a transformation of the plane itself where a line and its counterpart are parallel. From this definition, it easily follows that the dilations preserve angles, that is, they are conformal transformations of the plane, that the set of dilations form a 'group' and that translations are particular cases of dilations.

Let us consider the dilation in which the line OA becomes the line O'B, being O' the homologue of O and B the homologue of A. Necessarily, the lines OO' and AB are invariant in this dilation and the point H1, center of the dilation, is invariant. In this dilation, the circle with center O and radius OA becomes the circle with center O' and of radius O'B and the ratio of the dilation is the ratio (positive) of the segments O'B and OA.

If, on the contrary, point A becomes B' then the line AB' is invariant and the point H2 is the center of dilation. In this case, the ratio of dilation is negative.

Dilation axes

Given two circles, they can always be considered as homothetic to each other.

In the figure on the side, the lines of s1, are in the dilation of positive ratio, with center in P1, or of negative ratio, with center of dilation in N1.

Let us consider the dilations, one with center in P1 in which the circle S2 is homothetic of the circle s1, and the dilation of center P3 in which the circle s3 is homothetic to the circle s2. The composition of these two dilations is the dilation of center in P2 that transforms the circle s1 into the circle s3. It is for this reason that the positive dilation centers, P1, P2 and P3 are aligned. In general, given three circles there are six centers of homothety, aligned three by three on four straight lines.

These lines are the so-called axes of dilation of the three given circles.

References and their point