I have read over half a dozen different descriptions of Bézier splines, each emphasizing some different aspect of the subject. This description, however is by far the best of any of them. Watch it and weep! ❤️💓💕
... then watch this one on Bézier curves in projective geometry:
Here's Wildberger on the maths:
This gives an interesting way to look at the structure of the rational number field Q:
Here, the rationals on an interval are defined as a convex combination, just like Bézier splines. Here is a closer look at this structure, relating the Stern-Brocot tree to the Ford numbers and the Farey sequences.
Now look at this as a way of interpreting numbers, at 9 minutes 55 seconds, we can think of the number as an operator, with two different interconnected interpretations, according to whether we think of it being an operator in an additive group, or an operator in a multiplicative group. See the bit on fields and vector spaces in Wildberger on Chromogeometry. The operator is defined as a fixpoint function, which is effectively the fixpoint of the chain rule for differentiating composite functions.
To get soe ideas about how this could treat functions and their inverses, see Toby on Square Roots. Now see how calculus can be done algebraically on approximations of polynomial functions in a vector space:
The idea I am trying to sharpen up here, is that of modelling the procespro of finding numerical solutions to differential equations by a Cartesian Closed Category. So we would deal with any peoblem as ultimately being the solution of a fixpoint equation which sufficiently accurately approximates the underlying geometry as to produce a spectrum of solutions which are in some precise sense indistinguishable from one another. See Aristotle on The Continuum, All About e and Wildberger on Chromogeometry.
See this too
I think this is connected with Pontryagin's maximum principle, which is a central principle in optimal control theory developed by Lev Pontryagin and his students in the fifties, and which significantlt extends the theory of Lagrange to non-linear control systems. Furthermore, I think that it was Pontryagin's earlier theoretical work in groups and topological spaces which underlies this advance. See the essay mentioned in this post: Julia Galef on the Intuition Behind Bayes' Theorem.
What I hope to be able to produce as a result is a constructive model for proofs in Tensor Calculus. See Tensor Calculus. For the philosophy behind this idea, see Computation Considered as an Empirical Foundation ...
... then watch this one on Bézier curves in projective geometry:
Here's Wildberger on the maths:
This gives an interesting way to look at the structure of the rational number field Q:
Here, the rationals on an interval are defined as a convex combination, just like Bézier splines. Here is a closer look at this structure, relating the Stern-Brocot tree to the Ford numbers and the Farey sequences.
Now look at this as a way of interpreting numbers, at 9 minutes 55 seconds, we can think of the number as an operator, with two different interconnected interpretations, according to whether we think of it being an operator in an additive group, or an operator in a multiplicative group. See the bit on fields and vector spaces in Wildberger on Chromogeometry. The operator is defined as a fixpoint function, which is effectively the fixpoint of the chain rule for differentiating composite functions.
To get soe ideas about how this could treat functions and their inverses, see Toby on Square Roots. Now see how calculus can be done algebraically on approximations of polynomial functions in a vector space:
The idea I am trying to sharpen up here, is that of modelling the procespro of finding numerical solutions to differential equations by a Cartesian Closed Category. So we would deal with any peoblem as ultimately being the solution of a fixpoint equation which sufficiently accurately approximates the underlying geometry as to produce a spectrum of solutions which are in some precise sense indistinguishable from one another. See Aristotle on The Continuum, All About e and Wildberger on Chromogeometry.
See this too
I think this is connected with Pontryagin's maximum principle, which is a central principle in optimal control theory developed by Lev Pontryagin and his students in the fifties, and which significantlt extends the theory of Lagrange to non-linear control systems. Furthermore, I think that it was Pontryagin's earlier theoretical work in groups and topological spaces which underlies this advance. See the essay mentioned in this post: Julia Galef on the Intuition Behind Bayes' Theorem.
What I hope to be able to produce as a result is a constructive model for proofs in Tensor Calculus. See Tensor Calculus. For the philosophy behind this idea, see Computation Considered as an Empirical Foundation ...
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