This is a special path, the dividing line between those ellipses and that wavy line. Or setting it vertically there to start with and trusting no truck driving down the street will rattle it loose. That’s a very precise dot, where is exactly zero. These paths, the trajectories, match ...
where f is the vector with the entries of F in the row-major order. When A has rank 8, it is possible to obtain f (up to a scaling factor). Furthermore, when matrix A has rank 7, it is still possible to solve for f by using the singularity constraint, that is the constraint ...
A less complete, but still illustrative description depicts robots as general purpose motion machines. Motion is the topic of this and the next chapters. In other words, we focus our attention on the acting part.[111] At the same time, the concepts that you will learn here, together with ...
In other words, the union is itself Λ. And furthermore, (2) the union of the tower of sets Q0⊂Q1⊂Q2... is our desired position Q and that every member of Λ meets Q in exactly one point. The proof of this is below: (1) To see that the Λn tower is exhaustive, ...
One consequence of this theorem is that: If a curved surface is developed upon any other surface whatsoever, the measure of curvature in each point remains unchanged. In other words, surfaces with the same Gaussian curvature can be mapped into each other without any distortion. For example, sur...
In other words, K(x) is the norm of the point evaluation at x. \qquad Proof. \qquad s_{x}=\sup \left|\left(h, k_{x}\right)\right|^{2}=\left\|k_{x}\right\|^{2}=\left(k_{x}, k_{x}\right)=K(x). \square \qquad Much of the strength of the Bergman kernel comes...
It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in surveying, and its name is derived from Greek words meaning “Earth measurement.” Eventually it was realized that geometry need not be limited to the study of flat surfaces...
Then, for any two points p, q there is a path \gamma connecting them, such that L(\gamma)=\operatorname{dist}(p, q). In other words, the infimum in the definition of distance is always attained. Given a parametrization f:U→M with first fundamental form I, one can define the ...
which we also need, is part of the theory offewnomials, a precursor of tropical geometry: our argument improves Zilber’s in that it uses tropical geometry more systematically, together with some easy complex analysis, to weaken the conditions needed on an approximate solution to yield an actua...
to a fixed order type. without giving the precise definition, what theorem 1.1 states in this form is that the order type of \(g^*_n\) is projectively rigid with an error of o (1/ n ). or in other words, its “projective rigidity” is o (1/ n ), at least for large ...