Algebra and Geometry
Terms
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- What is n-dimensional Euclidean space, E^n?
- R^n with the distance function.
- What is d(x,y)?
- d(x,y)=|x-y|=((x-y).(x-y))^1/2
- Define a Euclidean isometry.
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A distance preserving transformation of E^n.
A function f:E^n->E^n s.t. d(f(x),f(y))=d(x,y) - Define a Euclidean transformation.
- Any transformation of the form fA,t:x->Ax+t where A is an nxn orthogonal matrix and t in E^n.
- fA,t o fB,u
- fAB,Au+t
- (fA,t)^-1
- fA^-1,-(A^-1)t
- What is the relationship between Euclidean isometries and Euclidean transformations?
- They are the same thing.
- Define a geodesic.
- A curve such that for any 2 points on the curve, the part of the curve between them is the shortest curve between them.