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seen Jul 26 at 8:39

It is impossible to disassociate language from science or science from language, because every natural science always involves three things: the sequence of phenomena on which the science is based, the abstract concepts which call these phenomena to mind, and the words in which the concepts are expressed. To call forth a concept, a word is needed; to portray a phenomenon, a concept is needed. All three mirror one and the same reality.

Antoine Lavoisier, 1789

[L]ogic, in so far as it exhibits the universal and necessary laws of the understanding, must in these very laws present us with criteria of truth. Whatever contradicts these rules is false, because thereby the understanding is made to contradict its own universal laws of thought; that is, to contradict itself.

Kant, Critique of Pure Reason, I, 2nd Part, II. Of Transcendental Logic

He who in reasoning cites authority is making use of his memory rather than of his intellect.

Leonardo Da Vinci, Thoughts on Art and Life


Mar
10
revised How are the Taylor Series derived?
added 87 characters in body
Mar
10
revised How are the Taylor Series derived?
added 82 characters in body
Mar
10
answered How are the Taylor Series derived?
Mar
6
awarded  Announcer
Feb
10
comment How elliptic arc can be represented by cubic Bézier curve?
+1. I wish I had seen this post earlier. I've recently implemented circular arcs in PostScript.
Sep
30
comment Surprising identities / equations
postscript source code for the image.
Sep
29
answered Surprising identities / equations
Sep
4
comment Intuition for Absorption and Distributive Laws in Elementary Logic
I suspect you may be throwing yourself off track with the "or all three" part in both of your attempts. For (AL1), there are only 2 terms.
Apr
14
answered Suggest an Antique Math Book worth reading?
Apr
8
comment Intersect a line with a bicubic Bezier Surface Patch.
Thanks. I'll check those out. I also stumbled across this: Implicitization and parametrization of quadratic and cubic surfaces by μ-bases which looks pertinent.
Apr
8
comment Intersect a line with a bicubic Bezier Surface Patch.
This really does appear to be the right answer both popularly and pragmatically, so I accept. I'm not giving up, but it's going to take me some time to understand Kajiya. First surprise is his use of homogeneous coordinates, so I'm going back to Maxwell for background, and I found a pdf of Modern Higher Algebra, by George Salmon, 1885, from wikipedia.
Apr
8
accepted Intersect a line with a bicubic Bezier Surface Patch.
Apr
4
comment Intersect a line with a bicubic Bezier Surface Patch.
I had a suggestion in comp.lang.postscript to rotate the whole space so the ray coincides with the x-axis. That appears to reduce it to $S_x(u,v)-t=0$, $S_y(u,v)=0$, and $S_z(u,v)=0$. Then I think I can use Kajiya's approach to solve for u and v separately. I don't really understand his presentation of Bezout's Theorem, but Wikipedia suggests I can skip that and use a Sylvester Matrix. ... I took a sample patch from the dataset and computed the algebraic terms. Even without doing a rotation, there are lots of zeros in the coefficients, which I find promising.
Apr
4
awarded  Commentator
Apr
4
comment Intersect a line with a bicubic Bezier Surface Patch.
Oh. I was looking at the wrong paper. SIGGRAPH 82. Got it now.
Apr
3
comment Intersect a line with a bicubic Bezier Surface Patch.
Thank you. It's a much stronger presentation, now. I know I'm being stubborn but I really think I can do something with these equations. I'll probably accept your answer in the end. But it would, in a sense, mean defeat, you know?
Apr
3
revised Intersect a line with a bicubic Bezier Surface Patch.
specified the type of surface more precisely in the title
Apr
2
comment Intersect a line with a bicubic Bezier Surface Patch.
+1 You've already helped me clean up my variable names.
Apr
2
revised Intersect a line with a bicubic Bezier Surface Patch.
made parameter names conform throughout
Apr
2
comment Intersect a line with a bicubic Bezier Surface Patch.
I've got a reprint of that paper in the IEEE Tutorial: Image Synthesis. But I have difficulty seeing where to plug-in the points. ... Everybody says they're nasty equations! But no one's published them.