Space coordinates or figures can move {transformation, space}.
translation
Motion can be along geodesic {translation, transformation}. Transformation shifts axes, keeping new axes parallel to old axes: x2 = x1 - h and y2 = y1 - k, where h and k are shift distances.
translation: shear
Translation can keep one coordinate axis or coordinate plane unchanged while others change {shear transformation} {shear translation}. Points move parallel to fixed axis or plane. Movement can be proportional to distance from fixed axis or plane.
dilation
Transformations {dilation transformation} can be through a fixed point, so distances from points to fixed point are a constant {constant of dilation} multiple of distances from fixed point to new points. Dilation is similar to similarity.
rotation
Rigid turning motion through angle can be around a common center or line {rotation, transformation}. Positive rotation is anti-clockwise. Negative rotation is clockwise. Transformation rotates both axes by angle A: x2 = x1*cos(A) + y1*sin(A) and y2 = - x1*sin(A) + y1*cos(A).
isometry
One-to-one transformation can leave distances, sizes, and shapes unchanged {isometry transformation}. Isometry involves translation and rotation. Rotate both axes by angle A and translate axes: x2 = x1*cos(A) + y1*sin(A) - h and y2 = - x1*sin(A) + y1*cos(A) - k, where h and k are shift distances.
reflection
Transformation can reflect both axes through origin: x2 = - x1 and y2 = - y2 {reflection, transformation}.
inversion
Transformation can involve both rotation and reflection {inversion, transformation}.
invariance
Transformations can result in same product as before {invariance, transformation}. Invariance example is geometric-figure rotation or reflection that transforms the points back into same figure {symmetry group, transformation}.
invariance: symmetry
For symmetric reflection through line or plane, if (x,y) is on figure, then (x,-y) or (-x,y) is on figure {axial symmetry transformation} {bilateral symmetry transformation}. For symmetric reflection through point, if (x,y) is on figure, then (-x,-y) is on figure {radial symmetry transformation} {point symmetry transformation}.
association
For three successive operations, find result of first and second and then do third, (a + b) + c, or do first then find result of second and third, a + (b + c) {association operation}. Results can be same product {associative}, (a + b) + c = a + (b + c), or different products {non-associative, transformation}, (a + b) + c != a + (b + c).
commutation
Two successive operations can happen in either order, a + b or b + a {commutation operation}. The result can be same product {commutative, transformation}, a + b = b + a, or different products {non-commutative, transformation}: a + b != b + a.
covariance
Transformations can result in same product as before but times constant.
contravariance
Transformations can result in same product as before, but using different coordinates.
group
Transformations form groups. Operations transform elements to other elements. In groups, transformations can reduce to other transformations or be irreducible. Metric determines possible coordinate transformations at manifold points. Transformations transform basis vectors into themselves, if transformation keeps same coordinate system. Transformations can transform basis vectors into their linear combinations.
Transformation groups can combine isometries, shear transformations, and similarities {affinities}.
Rational coordinate functions can have inverses {birational transformation}. Birational transforms can change irreducible algebraic plane curves to curves with no singular points, except for double points with distinct tangents.
If one figure derives from another by continuous changes, and derived figure has same generality as original figure, all first-figure properties are true of second figure {continuity principle} {correlativity principle} {contingent relations principle} {principle of continuity} {principle of correlativity} {principle of contingent relations}.
Derivatives of three-dimensional functions with spatial directions can be rational, single-valued, and solvable {Cremma transformation}. Cremma transforms can change irreducible algebraic plane curves to curves with no singular points, except for multiple points with distinct tangents.
Lines and points are duals in plane projective geometry. For plane figures, true theorems {reciprocal theorem} about non-metric properties can interchange the words "line" and "point" {duality}| {principle of duality}. For example, "a unique line intersects two points" and "a unique point intersects two lines".
three dimensions
Planes and points are duals in three-dimensional projective geometry. For plane figures, true theorems about non-metric properties can interchange the words "plane" and "point", if the word "line" does not change.
metric
These duality principles are not true for metric properties.
Plane points and lines can transform {homography transformation} into points and lines in the plane or another plane, to make homologous or projectively related figures.
Figures can move in space without shape or size distortion {superposition axiom}.
Point reflections make points {image, point}.
Rotations around symmetry center can result in coincidence. Rotations can be fractions of 360 degrees: 180, 120, 90, 72, 60, 45, 30, and 15 degrees. 360 divided by rotation degrees makes whole number {order of symmetry}: 2, 3, 4, 5, 6, 8, 12, and 24.
For symmetry through line or axis {axial symmetry}|, (x,y) goes to (x,-y) or (-x,y).
For symmetry through point or origin {radial symmetry}| {central symmetry} {point symmetry}, (x,y) goes to (x,-y), (-x,y), and (-x,-y).
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Date Modified: 2022.0225