Partial derivatives of surface curvature relative to a tangent plane












0












$begingroup$


If we take the tangent plane (say, $mathsf{T}$) to a convex surface $mathcal{S}$ associated with a given surface normal vector $mathtt{n}$, I assume the surface will have a fixed rate of change along a disk defined on $mathsf{T}$, centered at the base of $mathtt{n}$ (point $mathtt{p}$, say, considered as a point in $mathsf{T}$). For any $mathsf{T}$-direction emanating from $mathtt{p}$ $-$ perhaps designated as an angle $alpha$ from an arbitrarily selected $alpha = 0$ direction $-$ I assume the rate at which $mathcal{S}$ diverges from $mathsf{T}$ defines a continuous function (say, $varsigma$) on $alpha$ dependent on $mathcal{S}$ and $mathtt{n}$. The nature of $varsigma$ seems obvious for a few cases $-$ everywhere constant for a sphere; near-zero for an almost flat surface $-$ and I'd be interested in calculating (what I'm calling) $varsigma$ for other surfaces (maybe there's a recognized name for this function). For mesh geometry, I assume $varsigma$ can be calculated at any mesh point by examining the other mesh points connected to it, partitioning the disk around $mathtt{p}$ into sectors. But what I'm mostly interested in is the opposite direction $-$ is there a way to construct a triangulation/convex hull, or a NURBS surface, by asserting several normal vectors, and then for each normal present the desired $varsigma$ function either sampled at particular $alpha$s or as a closed function? Is this a recognized method for defining surfaces in Computer-Aided-Design?



Information I've found on this topic seems to address $mathcal{S}$ curvature as a field on curves in the neighborhood of $mathtt{p}$ (e.g. Geodesic Torsion) but I have not seen a statement of $varsigma$ as a one-place function on $alpha$ (for each $mathcal{S}$ and $mathtt{n}$). It seems as if this $varsigma$ might be a limit of the Geodesic Torsion on the space of all closed curves that wind once around $mathtt{p}$ in $mathtt{p}$-disks as their radius shrinks to zero $-$ or maybe equivalently just the torsion on circles of radius $rho$ as $rho rightarrow 0$ $-$ but maybe I'm picturing this wrong.










share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    If we take the tangent plane (say, $mathsf{T}$) to a convex surface $mathcal{S}$ associated with a given surface normal vector $mathtt{n}$, I assume the surface will have a fixed rate of change along a disk defined on $mathsf{T}$, centered at the base of $mathtt{n}$ (point $mathtt{p}$, say, considered as a point in $mathsf{T}$). For any $mathsf{T}$-direction emanating from $mathtt{p}$ $-$ perhaps designated as an angle $alpha$ from an arbitrarily selected $alpha = 0$ direction $-$ I assume the rate at which $mathcal{S}$ diverges from $mathsf{T}$ defines a continuous function (say, $varsigma$) on $alpha$ dependent on $mathcal{S}$ and $mathtt{n}$. The nature of $varsigma$ seems obvious for a few cases $-$ everywhere constant for a sphere; near-zero for an almost flat surface $-$ and I'd be interested in calculating (what I'm calling) $varsigma$ for other surfaces (maybe there's a recognized name for this function). For mesh geometry, I assume $varsigma$ can be calculated at any mesh point by examining the other mesh points connected to it, partitioning the disk around $mathtt{p}$ into sectors. But what I'm mostly interested in is the opposite direction $-$ is there a way to construct a triangulation/convex hull, or a NURBS surface, by asserting several normal vectors, and then for each normal present the desired $varsigma$ function either sampled at particular $alpha$s or as a closed function? Is this a recognized method for defining surfaces in Computer-Aided-Design?



    Information I've found on this topic seems to address $mathcal{S}$ curvature as a field on curves in the neighborhood of $mathtt{p}$ (e.g. Geodesic Torsion) but I have not seen a statement of $varsigma$ as a one-place function on $alpha$ (for each $mathcal{S}$ and $mathtt{n}$). It seems as if this $varsigma$ might be a limit of the Geodesic Torsion on the space of all closed curves that wind once around $mathtt{p}$ in $mathtt{p}$-disks as their radius shrinks to zero $-$ or maybe equivalently just the torsion on circles of radius $rho$ as $rho rightarrow 0$ $-$ but maybe I'm picturing this wrong.










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      If we take the tangent plane (say, $mathsf{T}$) to a convex surface $mathcal{S}$ associated with a given surface normal vector $mathtt{n}$, I assume the surface will have a fixed rate of change along a disk defined on $mathsf{T}$, centered at the base of $mathtt{n}$ (point $mathtt{p}$, say, considered as a point in $mathsf{T}$). For any $mathsf{T}$-direction emanating from $mathtt{p}$ $-$ perhaps designated as an angle $alpha$ from an arbitrarily selected $alpha = 0$ direction $-$ I assume the rate at which $mathcal{S}$ diverges from $mathsf{T}$ defines a continuous function (say, $varsigma$) on $alpha$ dependent on $mathcal{S}$ and $mathtt{n}$. The nature of $varsigma$ seems obvious for a few cases $-$ everywhere constant for a sphere; near-zero for an almost flat surface $-$ and I'd be interested in calculating (what I'm calling) $varsigma$ for other surfaces (maybe there's a recognized name for this function). For mesh geometry, I assume $varsigma$ can be calculated at any mesh point by examining the other mesh points connected to it, partitioning the disk around $mathtt{p}$ into sectors. But what I'm mostly interested in is the opposite direction $-$ is there a way to construct a triangulation/convex hull, or a NURBS surface, by asserting several normal vectors, and then for each normal present the desired $varsigma$ function either sampled at particular $alpha$s or as a closed function? Is this a recognized method for defining surfaces in Computer-Aided-Design?



      Information I've found on this topic seems to address $mathcal{S}$ curvature as a field on curves in the neighborhood of $mathtt{p}$ (e.g. Geodesic Torsion) but I have not seen a statement of $varsigma$ as a one-place function on $alpha$ (for each $mathcal{S}$ and $mathtt{n}$). It seems as if this $varsigma$ might be a limit of the Geodesic Torsion on the space of all closed curves that wind once around $mathtt{p}$ in $mathtt{p}$-disks as their radius shrinks to zero $-$ or maybe equivalently just the torsion on circles of radius $rho$ as $rho rightarrow 0$ $-$ but maybe I'm picturing this wrong.










      share|cite|improve this question









      $endgroup$




      If we take the tangent plane (say, $mathsf{T}$) to a convex surface $mathcal{S}$ associated with a given surface normal vector $mathtt{n}$, I assume the surface will have a fixed rate of change along a disk defined on $mathsf{T}$, centered at the base of $mathtt{n}$ (point $mathtt{p}$, say, considered as a point in $mathsf{T}$). For any $mathsf{T}$-direction emanating from $mathtt{p}$ $-$ perhaps designated as an angle $alpha$ from an arbitrarily selected $alpha = 0$ direction $-$ I assume the rate at which $mathcal{S}$ diverges from $mathsf{T}$ defines a continuous function (say, $varsigma$) on $alpha$ dependent on $mathcal{S}$ and $mathtt{n}$. The nature of $varsigma$ seems obvious for a few cases $-$ everywhere constant for a sphere; near-zero for an almost flat surface $-$ and I'd be interested in calculating (what I'm calling) $varsigma$ for other surfaces (maybe there's a recognized name for this function). For mesh geometry, I assume $varsigma$ can be calculated at any mesh point by examining the other mesh points connected to it, partitioning the disk around $mathtt{p}$ into sectors. But what I'm mostly interested in is the opposite direction $-$ is there a way to construct a triangulation/convex hull, or a NURBS surface, by asserting several normal vectors, and then for each normal present the desired $varsigma$ function either sampled at particular $alpha$s or as a closed function? Is this a recognized method for defining surfaces in Computer-Aided-Design?



      Information I've found on this topic seems to address $mathcal{S}$ curvature as a field on curves in the neighborhood of $mathtt{p}$ (e.g. Geodesic Torsion) but I have not seen a statement of $varsigma$ as a one-place function on $alpha$ (for each $mathcal{S}$ and $mathtt{n}$). It seems as if this $varsigma$ might be a limit of the Geodesic Torsion on the space of all closed curves that wind once around $mathtt{p}$ in $mathtt{p}$-disks as their radius shrinks to zero $-$ or maybe equivalently just the torsion on circles of radius $rho$ as $rho rightarrow 0$ $-$ but maybe I'm picturing this wrong.







      differential-geometry partial-derivative 3d surfaces computer-vision






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 25 '18 at 15:15









      Nathaniel ChristenNathaniel Christen

      11




      11






















          0






          active

          oldest

          votes












          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "69"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3052180%2fpartial-derivatives-of-surface-curvature-relative-to-a-tangent-plane%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          0






          active

          oldest

          votes








          0






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes
















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3052180%2fpartial-derivatives-of-surface-curvature-relative-to-a-tangent-plane%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          Bundesstraße 106

          Verónica Boquete

          Ida-Boy-Ed-Garten