Further, a conical surface is invariant as a set under a homothety of space.

The positive homothety ratio is at most 2 and:

Any homothety is an isometry of , in particular including the map:

Thus homothety is added, self-similarity is considered a symmetry.

Homothetic or Homothety may refer to:

If the two tangent circle touch collinear pairs of antihomologous point - as in Figure 5 - then because of the homothety.

Some other names for dilation are uniform scaling, isotropic scaling, homothety, and homothecy.

(homothety and rotation)

Homothety, interactive applet from Cut-the-Knot.

In this case, the geometrical similarity becomes an homothety and the thermal center is the center of the homothety itself.

Property 3 in the formulation of the homothety immediately follows; the triangle of points of tangency is known as the anticomplementary triangle.

It is easily proven that triangles OPQ and OP'Q' are similar because of the homothety.

The medial triangle can also be viewed as the image of triangle ABC transformed by a homothety centered at the centroid with ratio -1/2.

A Fubini-Study metric is determined up to homothety (overall scaling) by invariance under such a U(n+1) action; thus it is homogeneous.

Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence.

Since such a homothety is a congruence, this gives property 5, and also the Johnson circles theorem since congruent triangles have circumscribed circles of equal radius.

A homothetic vector field (sometimes homothetic collineation or homothety) is a projective vector field which satisfies the condition:

Here d(S) is the pushforward along the scalar homothety A particular case of a non-linear connection arising in this manner is that associated to a Finsler manifold.

Uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety or dilation.

Now the midpoints of the sides of any triangle are the images of its vertices by a homothety with factor , centered at the barycenter of the triangle.

Sometimes, by extension, a surface is called a Clifton-Pohl torus if it is a finite covering of the quotient of by any homothety of ratio different from .

As a consequence, in the specific case in which S O, the homothety becomes a linear transformation, which preserves not only the collinearity of points (straight lines are mapped to straight lines), but also vector addition and scalar multiplication.

The TC may exist even if the body has non-homogeneous isotropic thermal properties, but when this condition is not verified, it's not possible to determine its position by using the simple geometric method shown above, and the transformation will not be a homothety.

Applied to the anticomplementary triangle, which is itself obtained from the Johnson triangle by a homothety with factor 2, it follows from composition of homotheties that the reference triangle is homothetic to the Johnson triangle by a factor 1.

For every convex body C in the plane, we can inscribe a rectangle r in C such that a homothetic copy R of r is circumscribed about C and the positive homothety ratio is at most 2 and .

Any homothecy centered in the origin of a vector space, where c is a scalar, is a linear operator.

Some other names for dilation are uniform scaling, isotropic scaling, homothety, and homothecy.

The composition of the three then is an element of the group of homothecy-translations that fixes B, so it is a homothecy with center B, possibly with ratio 1 (in which case it is the identity).

Geometrically, a diagonalizable matrix is an inhomogeneous dilation (or anisotropic scaling) - it scales the space, as does a homogeneous dilation, but by a different factor in each direction, determined by the scale factors on each axis (diagonal entries).