The direction space of Euclidean is
On the direction space of , one can also define the โdistanceโ between any two points : by restricting the โmetricโ of to , we obtain the metric of , and then using the metric of , we can define the distance on as the length of the shortest geodesic connecting the two points
Maps that preserve the metric of are called isometries of . It can be proven that isometries of are affine maps. The (orientation-preserving) isometries of are . The part of the isometries that fixes the origin is , which also preserves the direction space of and preserves the metric on , thereby preserving the distance on
is called rotation
Elements of are . Set-theoretically equivalent to
The action of on and on is compatible
Thus, we can define multiplication on as the corresponding multiplication of
At the same time, because are linear maps, the distributive law holds
[complex_numbler_geometric_meaning] This is equivalent to the multiplication of unit complex numbers. Since rotations and scalar multiplication commute, the multiplication on unit complex numbers can be easily extended to multiplication on complex numbers
[angle] Angle
It may not be a perfect motivation
Intuitively, in Euclidean , we can โrotateโ, and the composition of rotations corresponds to the addition of โanglesโ
Angle should be the distance on , the distance along the shortest geodesic connecting two points.
The rotation we defined is the group preserving the distance, which hasnโt directly explained the intuitive sense of rotation.
Intuitively, we know that on , we can continuously define a โpositive direction of the tangent spaceโ for each point
For , the positive direction of the tangent space is
One can guess that the intuitive rotation is moving each point on along the positive direction by a distance/radian/angle of
The question is, how to understand that , which preserves the distance (and orientation), is equivalent to this intuitive rotation?
At least from the result, I know
- The positive direction tangent field is an infinitesimal isometry (called a Killing field), which is the Lie algebra of the isometry group. And moving each point along the positive direction by a distance (along the geodesic) is the way to generate an isometry from an infinitesimal isometry, i.e., the of the Lie algebra.
-
Since geodesics are a kind of integral curve along a vector field, the addition of distances is homomorphic to the multiplication of group actions
This also gives commutativity
For the geodesic coordinates starting from , the result is denoted as [trigonometric_function] trigonometric functions .
According to the correspondence between multiplication and multiplication, we know that
Therefore is also the action of rotating other points on by
According to the homomorphism
According to the power series representation of , and that the unit tangent vector in the positive direction at is corresponding to
we can obtain
Expressed using complex numbers,
and we have
or or
Complex conjugation preserves distance but reverses direction or
Similarly, hyperbolic and split complex
-
give
-
give
The definitions of are analogous to those of trigonometric functions and angles in . Restricting to yields a Euclidean-type metric manifold (Riemannian manifold). Starting from , moving along a geodesic by yields , and projecting onto the axes gives the hyperbolic cosine and hyperbolic sine .
Connected to split-complex via .
. Distance can also be expressed in . .
[hyperbolic_exp_inverse]
is monotonically increasing.
Solving the quadratic equation yields the inverse mapping.
inverse
Question Generalize to quaternions and octonions and their split versions
Conversely, if we accept the concept that angles are additive (homomorphism), then it provides motivation for: the Euclidean norm of being and not some other norm. (Norm is defined as .)