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On curves with constant curvatures

Y.M. Ebbens, (2015) On curves with constant curvatures. Bachelor's Thesis, Mathematics.

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Abstract

In Euclidean geometry a curve is determined completely by its curvatures, up to a rigid transformation. Affine geometry, which studies geometric invariants under the group of volume preserving linear transformations, gives rise to the so called affine curvatures of a curve. In my thesis I will first derive two necessary and sufficient conditions for a curve to have constant Euclidean curvatures. Firstly, the tangents at any two points make the same angle with the line segment connecting these points. Secondly, the distance between points on the curve does not depend on the actual position of these points, but only on the arc length of the curve segment between the points. Then I will derive similar conditions for a curve to have constant affine curvatures. Firstly, the volume of a full-dimensional simplex formed by points on the curve does not depend on the actual positions of these points, but only on the affine arc length of the curve segments between the points. Secondly, the affine arc length of an off-set curve in the tangential direction is proportional to the affine arc length of the curve itself.

Item Type: Thesis (Bachelor's Thesis)
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 15 Feb 2018 08:04
Last Modified: 15 Feb 2018 08:04
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/12688

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