In the case of plane or spatial coordinates, datum transformation takes the form of a similarity or Helmert transformation, consisting of a rotation and scaling operation in addition to a simple translation.

If these conditions do not hold, the formula describes a more general affine transformation of the plane provided that the determinant of A is not zero.

A further conformal transformation from the sphere to the plane gives a conformal double projection from the ellipsoid to the plane.

For example, if the affine transformation acts on the plane and if the determinant of A is 1 or − 1 then the transformation is an equi-areal mapping.

Such transformations form a subgroup called the equi-affine group A transformation that is both equi-affine and a similarity is an isometry of the plane taken with Euclidean distance.

To visualise the general affine transformation of the Euclidean plane, take labelled parallelograms ABCD and A ′ B ′ C ′ D ′.

Whatever the choices of points, there is an affine transformation T of the plane taking A to A ′, and each vertex similarly.

The dual number plane is used to represent the naive spacetime of Galileo in a study called Galilean invariance since the classical event transformation with velocity v looks like:

A map of the extended complex plane ( which is conformally equivalent to a sphere ) onto itself is conformal if and only if it is a Möbius transformation.

If =, then such transformation is known as point reflection, which on the plane ( = 2 ) is the same as the half-turn ( 180 °) rotation ; see below.

For any affine transformation of the Gaussian plane, z mapping to a z + b, a ≠ 0, a triangle is transformed but does not change its shape.

We want to find n ′ perpendicular to P. Let t be a vector on the tangent plane and M < sub > l </ sub > be the upper 3x3 matrix ( translation part of transformation does not apply to normal or tangent vectors ).

), various toolpaths ( top construction and tool planes only ), toolpath transformation in top plane, backplot, verify, posting.

The stages of cell plate formation include ( 1 ) creation of the phragmoplast, an array of microtubules that guides and supports the formation of the cell plate ; ( 2 ) trafficking of vesicles to the division plane and their fusion to generate a tubular-vesicular network ; ( 3 ) continued fusion of membrane tubules and their transformation into membrane sheets upon the deposition of callose, followed by deposition of cellulose and other cell wall components ; ( 4 ) recycling of excess membrane and other material from the cell plate ; and ( 5 ) fusion with the parental cell wall

It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface.

In 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to the axis.

The Haybailer's transformation from plane to car seems to have significantly weakened its flying ability, and Max often has to bail out when the Haybailer breaks down.

In lower bainite, the habit plane in ferrite will also shift from < 111 > towards < 110 > as transformation temperature decreases.

In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion.

Word of Dag Hammarskjold's death in an African plane crash has sent a shockwave around the globe.

Hence, thought of as a line in a particular plane **yp, any tangent to Q has a unique image and moreover this image is the same for all planes through L.

For any pencil in a plane containing a Af-fold secant of **zg has an image regulus which meets the plane of the pencil in Af lines, namely the images of the lines of the pencil which pass through the intersection of **zg and the multiple secant, plus an additional component to account for the intersections of the images of the general lines of the pencil.

`` For three days now a German reconnaissance plane has been over the city taking pictures.

This is not only a compliment to Mijbil, of whom there are a fine series of photographs and drawings in the book, but to the author who has catalogued the saga of a frightened otter cub's journey by plane from Iraq to London, then by train ( where he lay curled in the wash basin playing with the water tap ) to Camusfearna, with affectionate detail.

Members of the family are usually perennial herbs with sword-shaped unifacial leaves ; the inflorescence is a spike or panicle of solitary flowers, or forms a monochasial cyme or rhipidium ( meaning that the successive stems of the flowers follow a zig-zag path in the same plane ); and the flower has only three stamens, each opposite to an outer tepal.

In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates.

All that at present can be attempted is, to reproduce a single plane in another plane ; but even this has not been altogether satisfactorily accomplished, aberrations always occur, and it is improbable that these will ever be entirely corrected.

On the one hand, Bernard Bolzano ( 1817 ) had been aware that any bounded sequence of points ( in the line or plane, for instance ) has a subsequence that must eventually get arbitrarily close to some other point, called a limit point.

If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable: in this case the system transfer function has non-repeated poles at complex plane origin ( i. e. their real and complex component is zero in the continuous time case ).

Crochet has been used by mathematician Daina Taimina in order to create a version of the hyperbolic plane.

The center of an atomic projectile striking this plane has geometrically a definite probability of passing within a certain distance of one of these points.

The rotation itself is represented by the angular velocity vector Ω, which is normal to the plane of the orbit ( using the right-hand rule ) and has magnitude given by:

* a triangle: A plane figure that has 3 straight bounding sides.

* a quadrilateral: A plane figure that has 4 straight bounding sides.

A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it ( for example, to locate a point on the surface of a sphere you need both its latitude and its longitude ).

Likewise, if he / she chooses to observe the interference pattern, then he / she has no influence whatsoever over where in the observation plane he / she will observe a specific particle.

* In the plane ( d = 2 ), if there are b vertices on the convex hull, then any triangulation of the points has at most 2n − 2 − b triangles, plus one exterior face ( see Euler characteristic ).

* In the plane, each vertex has on average six surrounding triangles.

Many results about plane figures are proved, e. g., If a triangle has two equal angles, then the sides subtended by the angles are equal.

Once the Euclidean plane has been described in this language, it is actually a simple matter to extend its concept to arbitrary dimensions.

If any part of the ball reaches any part of the imaginary vertical plane transected by this line while in-bounds and in possession of a player whose team is striving toward that end of the field, this is considered a touchdown and scores six points for the team whose player has advanced the ball to, or recovered the ball in, this position.