Octahedron Pyramid











up vote
0
down vote

favorite












octahedron



So, each octahedron can be inscribed in a cube, so that
the corner points of the octahedron are in the midpoints of the side areas of the cube, am I right?



From the octahedron $ABCDS_1S_2$, shown in the image, the vertices



$A =(13 | -5 | 3)$



, $B =(11 | 3 | 1)$



, $C =(5 | 3 | 7)$



and



$S_1 =(13 | 1 | 9)$



are given.
This octahedron is inscribed in the illustrated cube with the corners $P_1$ to $P_8$.



Now let



$E_0: 2x_1 + x_2 + 2x_3 + 9 * (2a-5) = 0, a ∈ ℝ$



be a set of planes $aEa$; Let $h$ be the line passing through the points $S_1$ and
$S_2 =(5 | -3 | 1).$





Now the task:
For $0 <a ≤ 1$, the plane $E_a$ intersects a pyramid of the octahedron
with the peak $S_1$:



Octahedron with Pyramid



I have to find the point of intersection $P_a$ of the plane $E_a$ with the line $h$ and then the volume $V_a$ of the truncated pyramid.





This is what I've done:
I have determined



$P_a=(13-4a|1-2a|9-4a)$



but how can I find the volume?
If anyone needs the math for determining $P_a$, please say so and I'll add my calculations.



Thx










share|cite|improve this question




























    up vote
    0
    down vote

    favorite












    octahedron



    So, each octahedron can be inscribed in a cube, so that
    the corner points of the octahedron are in the midpoints of the side areas of the cube, am I right?



    From the octahedron $ABCDS_1S_2$, shown in the image, the vertices



    $A =(13 | -5 | 3)$



    , $B =(11 | 3 | 1)$



    , $C =(5 | 3 | 7)$



    and



    $S_1 =(13 | 1 | 9)$



    are given.
    This octahedron is inscribed in the illustrated cube with the corners $P_1$ to $P_8$.



    Now let



    $E_0: 2x_1 + x_2 + 2x_3 + 9 * (2a-5) = 0, a ∈ ℝ$



    be a set of planes $aEa$; Let $h$ be the line passing through the points $S_1$ and
    $S_2 =(5 | -3 | 1).$





    Now the task:
    For $0 <a ≤ 1$, the plane $E_a$ intersects a pyramid of the octahedron
    with the peak $S_1$:



    Octahedron with Pyramid



    I have to find the point of intersection $P_a$ of the plane $E_a$ with the line $h$ and then the volume $V_a$ of the truncated pyramid.





    This is what I've done:
    I have determined



    $P_a=(13-4a|1-2a|9-4a)$



    but how can I find the volume?
    If anyone needs the math for determining $P_a$, please say so and I'll add my calculations.



    Thx










    share|cite|improve this question


























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      octahedron



      So, each octahedron can be inscribed in a cube, so that
      the corner points of the octahedron are in the midpoints of the side areas of the cube, am I right?



      From the octahedron $ABCDS_1S_2$, shown in the image, the vertices



      $A =(13 | -5 | 3)$



      , $B =(11 | 3 | 1)$



      , $C =(5 | 3 | 7)$



      and



      $S_1 =(13 | 1 | 9)$



      are given.
      This octahedron is inscribed in the illustrated cube with the corners $P_1$ to $P_8$.



      Now let



      $E_0: 2x_1 + x_2 + 2x_3 + 9 * (2a-5) = 0, a ∈ ℝ$



      be a set of planes $aEa$; Let $h$ be the line passing through the points $S_1$ and
      $S_2 =(5 | -3 | 1).$





      Now the task:
      For $0 <a ≤ 1$, the plane $E_a$ intersects a pyramid of the octahedron
      with the peak $S_1$:



      Octahedron with Pyramid



      I have to find the point of intersection $P_a$ of the plane $E_a$ with the line $h$ and then the volume $V_a$ of the truncated pyramid.





      This is what I've done:
      I have determined



      $P_a=(13-4a|1-2a|9-4a)$



      but how can I find the volume?
      If anyone needs the math for determining $P_a$, please say so and I'll add my calculations.



      Thx










      share|cite|improve this question















      octahedron



      So, each octahedron can be inscribed in a cube, so that
      the corner points of the octahedron are in the midpoints of the side areas of the cube, am I right?



      From the octahedron $ABCDS_1S_2$, shown in the image, the vertices



      $A =(13 | -5 | 3)$



      , $B =(11 | 3 | 1)$



      , $C =(5 | 3 | 7)$



      and



      $S_1 =(13 | 1 | 9)$



      are given.
      This octahedron is inscribed in the illustrated cube with the corners $P_1$ to $P_8$.



      Now let



      $E_0: 2x_1 + x_2 + 2x_3 + 9 * (2a-5) = 0, a ∈ ℝ$



      be a set of planes $aEa$; Let $h$ be the line passing through the points $S_1$ and
      $S_2 =(5 | -3 | 1).$





      Now the task:
      For $0 <a ≤ 1$, the plane $E_a$ intersects a pyramid of the octahedron
      with the peak $S_1$:



      Octahedron with Pyramid



      I have to find the point of intersection $P_a$ of the plane $E_a$ with the line $h$ and then the volume $V_a$ of the truncated pyramid.





      This is what I've done:
      I have determined



      $P_a=(13-4a|1-2a|9-4a)$



      but how can I find the volume?
      If anyone needs the math for determining $P_a$, please say so and I'll add my calculations.



      Thx







      euclidean-geometry vector-analysis plane-curves






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 18 at 15:17

























      asked Nov 18 at 13:02









      calculatormathematical

      389




      389






















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          1
          down vote



          accepted










          For heaven's sake write points data as $A=(13,-5,9)$, and so on!$quad$ [Note that $A=(13,-5,9)$ is a proposition, while $A(13,-5,9)$ is a function value.]



          One finds that the center of the octahedron $O$ is at $M=(9,-1,5)$, so that $vec{MS_1}=(4,2,4)$, which is orthogonal to the planes $2x_1+x_2+2x_3={rm const.}$ It follows that all these planes intersect the axis $S_1vee S_2$ of the octahedron orthogonally.



          You have obtained $P_a=(13-4a,1-2a,9-4a)$ [I have not checked this], so that
          $P_0=S_1$ and $P_1=M$. This allows to conclude that for $0< a<1$ the plane $E_a$ cuts off a small square pyramid $Y_a$ from $O$ with apex at $S_1$. The volume of this pyramid can be computed using elementary geometry. Note that the edge length $s$ of $O$ satisfies $s^2=|AB|^2=72$, and the height of the "upper half" $Y_1$ of $O$ is given by $h=|MS_1|=6$. It follows that ${rm vol}(Y_1)={1over3}s^2 h=144$. Since each $Y_a$ is similar to $Y_1$ with a linear factor $a$ we finally obtain
          $${rm vol}(Y_a)=144a^3 .$$






          share|cite|improve this answer





















            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',
            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%2f3003500%2foctahedron-pyramid%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            1
            down vote



            accepted










            For heaven's sake write points data as $A=(13,-5,9)$, and so on!$quad$ [Note that $A=(13,-5,9)$ is a proposition, while $A(13,-5,9)$ is a function value.]



            One finds that the center of the octahedron $O$ is at $M=(9,-1,5)$, so that $vec{MS_1}=(4,2,4)$, which is orthogonal to the planes $2x_1+x_2+2x_3={rm const.}$ It follows that all these planes intersect the axis $S_1vee S_2$ of the octahedron orthogonally.



            You have obtained $P_a=(13-4a,1-2a,9-4a)$ [I have not checked this], so that
            $P_0=S_1$ and $P_1=M$. This allows to conclude that for $0< a<1$ the plane $E_a$ cuts off a small square pyramid $Y_a$ from $O$ with apex at $S_1$. The volume of this pyramid can be computed using elementary geometry. Note that the edge length $s$ of $O$ satisfies $s^2=|AB|^2=72$, and the height of the "upper half" $Y_1$ of $O$ is given by $h=|MS_1|=6$. It follows that ${rm vol}(Y_1)={1over3}s^2 h=144$. Since each $Y_a$ is similar to $Y_1$ with a linear factor $a$ we finally obtain
            $${rm vol}(Y_a)=144a^3 .$$






            share|cite|improve this answer

























              up vote
              1
              down vote



              accepted










              For heaven's sake write points data as $A=(13,-5,9)$, and so on!$quad$ [Note that $A=(13,-5,9)$ is a proposition, while $A(13,-5,9)$ is a function value.]



              One finds that the center of the octahedron $O$ is at $M=(9,-1,5)$, so that $vec{MS_1}=(4,2,4)$, which is orthogonal to the planes $2x_1+x_2+2x_3={rm const.}$ It follows that all these planes intersect the axis $S_1vee S_2$ of the octahedron orthogonally.



              You have obtained $P_a=(13-4a,1-2a,9-4a)$ [I have not checked this], so that
              $P_0=S_1$ and $P_1=M$. This allows to conclude that for $0< a<1$ the plane $E_a$ cuts off a small square pyramid $Y_a$ from $O$ with apex at $S_1$. The volume of this pyramid can be computed using elementary geometry. Note that the edge length $s$ of $O$ satisfies $s^2=|AB|^2=72$, and the height of the "upper half" $Y_1$ of $O$ is given by $h=|MS_1|=6$. It follows that ${rm vol}(Y_1)={1over3}s^2 h=144$. Since each $Y_a$ is similar to $Y_1$ with a linear factor $a$ we finally obtain
              $${rm vol}(Y_a)=144a^3 .$$






              share|cite|improve this answer























                up vote
                1
                down vote



                accepted







                up vote
                1
                down vote



                accepted






                For heaven's sake write points data as $A=(13,-5,9)$, and so on!$quad$ [Note that $A=(13,-5,9)$ is a proposition, while $A(13,-5,9)$ is a function value.]



                One finds that the center of the octahedron $O$ is at $M=(9,-1,5)$, so that $vec{MS_1}=(4,2,4)$, which is orthogonal to the planes $2x_1+x_2+2x_3={rm const.}$ It follows that all these planes intersect the axis $S_1vee S_2$ of the octahedron orthogonally.



                You have obtained $P_a=(13-4a,1-2a,9-4a)$ [I have not checked this], so that
                $P_0=S_1$ and $P_1=M$. This allows to conclude that for $0< a<1$ the plane $E_a$ cuts off a small square pyramid $Y_a$ from $O$ with apex at $S_1$. The volume of this pyramid can be computed using elementary geometry. Note that the edge length $s$ of $O$ satisfies $s^2=|AB|^2=72$, and the height of the "upper half" $Y_1$ of $O$ is given by $h=|MS_1|=6$. It follows that ${rm vol}(Y_1)={1over3}s^2 h=144$. Since each $Y_a$ is similar to $Y_1$ with a linear factor $a$ we finally obtain
                $${rm vol}(Y_a)=144a^3 .$$






                share|cite|improve this answer












                For heaven's sake write points data as $A=(13,-5,9)$, and so on!$quad$ [Note that $A=(13,-5,9)$ is a proposition, while $A(13,-5,9)$ is a function value.]



                One finds that the center of the octahedron $O$ is at $M=(9,-1,5)$, so that $vec{MS_1}=(4,2,4)$, which is orthogonal to the planes $2x_1+x_2+2x_3={rm const.}$ It follows that all these planes intersect the axis $S_1vee S_2$ of the octahedron orthogonally.



                You have obtained $P_a=(13-4a,1-2a,9-4a)$ [I have not checked this], so that
                $P_0=S_1$ and $P_1=M$. This allows to conclude that for $0< a<1$ the plane $E_a$ cuts off a small square pyramid $Y_a$ from $O$ with apex at $S_1$. The volume of this pyramid can be computed using elementary geometry. Note that the edge length $s$ of $O$ satisfies $s^2=|AB|^2=72$, and the height of the "upper half" $Y_1$ of $O$ is given by $h=|MS_1|=6$. It follows that ${rm vol}(Y_1)={1over3}s^2 h=144$. Since each $Y_a$ is similar to $Y_1$ with a linear factor $a$ we finally obtain
                $${rm vol}(Y_a)=144a^3 .$$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 18 at 14:10









                Christian Blatter

                171k7111325




                171k7111325






























                    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.





                    Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                    Please pay close attention to the following guidance:


                    • 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.


                    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%2f3003500%2foctahedron-pyramid%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

                    AnyDesk - Fatal Program Failure

                    How to calibrate 16:9 built-in touch-screen to a 4:3 resolution?

                    QoS: MAC-Priority for clients behind a repeater