In Euclidean geometry, oblique reflections generalize ordinary reflections by not requiring that reflection be done using perpendiculars.If two points are oblique reflections of each other, they will still stay so under affine transformations.Consider a plane P in the three-dimensional Euclidean space.…

Also, sets of parallel lines remain parallel after an affine transformation.An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.…

The larger images can then be searched in a small window around the start position to find the best template location.Other methods can handle problems such as translation, scale, image rotation and even all affine transformations.Improvements can be made to the matching method by using more than one template (eigenspaces), these other templates can have different scales and rotations.…

Methods of drawing an ellipse usually require the axes of the ellipse to be known.An ellipse can be seen as an image of the unit circle under an affine transformation.…

In a third paper, Shi and Tomasi proposed an additional stage of verifying that features were tracked correctly.An affine transformation is fit between the image of the currently tracked feature and its image from a non-consecutive previous frame.If the affine compensated image is too dissimilar the feature is dropped.…

A number of analysis tools exist based on linear models, such as harmonic analysis, and they can all be used in neural networks with this linear neuron.The bias term allows us to make affine transformations to the data.…

Under repeated iteration of the horseshoe map, most orbits end up at the fixed point in the left cap.This is because the horseshoe maps the left cap into itself by an affine transformation that has exactly one fixed point.Any orbit that lands on the left cap never leaves it and converges to the fixed point in the left cap under iteration.…

This transformation skews and translates the original triangle.In fact, all triangles are related to one another by affine transformations.This is also true for all parallelograms, but not for all quadrilaterals.…

Each of the leaves of the fern is related to each other leaf by an affine transformation.For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation.…

The fact that affine transformations preserve equidissections also means that certain results can be easily generalized.All results stated for a regular polygon also hold for affine-regular polygons; in particular, results concerning the unit square also apply to other parallelograms, including rectangles and rhombuses.…

Since an affine transformation preserves straight lines and ratios of areas, it sends equidissections to equidissections.This means that one is free to apply any affine transformation to a polygon that might give it a more manageable form.…

It is important to note that each of the scores have diffent magnitudes and locations.The magnitude differences are not relevant however as scores remain proper under affine transformation.Therefore, to compare different score it is necessary to move them to a common scale.…

This provides a way to identify the tangent planes at different points along the curve: in particular, a tangent vector in the tangent space at one point on the curve is identified with a unique tangent vector at any other point on the curve.These identifications are always given by affine transformations from one tangent plane to another.…

Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations.These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwise invariant.…

A translation is an affine transformation with no fixed points.Matrix multiplications always have the origin as a fixed point.…

The set of executions of a set of statements within a possibly imperfectly nested set of loops is seen as the union of a set of polytopes representing the executions of the statements.Affine transformations are applied to these polytopes, producing a description of a new execution order.The boundaries of the polytopes, the data dependencies, and the transformations are often described using systems of constraints, and this approach is often referred to as a constraint-based approach to loop optimization.…

These points form triangles of four different shapes, with minimum area 1/8, as large as possible for six points in the square.This solution is an affine transformation of a regular hexagon but larger numbers of points have solutions that include interior points of the square.In discrete geometry and discrepancy theory, the Heilbronn triangle problem is a problem of placing points within a region in the plane, in order to avoid triangles of small area.…

This was done by minimizing the correlation between linear transformations of input/output bits, and at the same time minimizing the difference propagation probability.In addition, to strengthen the S-Box against algebraic attacks, the affine transformation was added.In the case of suspicion of a backdoor being built into the cipher, the current S-box might be replaced by another one.…

This range is implied by the fact that the mean lies within one standard deviation of any median.Under an affine transformation of the variable (X), the value of S does not change except for a possible change in sign.…

Self concordance is preserved under addition, affine transformations, and scalar multiplication by a value greater than one.Among other things, self-concordant functions are useful in the analysis of Newton's method.…