Motivation for 2D homogeneous coordinates: ratio always determinate












0














I'm working through an old textbook called Algebraic Projective Geometry, by Semple and Kneebone.



Early in the text, the authors write:




When we introduced complex points (on p. 12) we explained that they are to be regarded as ideal points adjoined to the euclidean plane; and the same is now true of points at infinity. These too are ideal points, but since they have not been introduced by way of coordinate representation it is not immediately obvious how they can be handled algebraically. This can, in fact, be managed quite easily, as we shall now show, by making the coordinates homogeneous.



When a pair of axes has been chosen, all actual (i.e. non-ideal points of the plane admit of unique and unexceptional representation by pairs of real numbers (X, Y). Two lines, whose equations are



aX + bY + c = 0 and a'X + b'Y + c' = 0,



meet, if they are not parallel, in the actual point whose coordinates are given by



X : Y : 1 = bc' - b'c : ca' - c'a : ab' - a'b.



Now whether or not the lines are parallel, supposing only that they are distinct, the ratios



bc' - b'c : ca' - c'a : ab' - a'b



are always determinate, in the sense that at least one of the quantities concerned is different from zero. Our object, then, must be to represent points by such sets of ratios rather than by pairs of numbers which are liable to become infinite.




I don't understand how the ratios are always determinate. I can't see that at least one of the quantities concerned is different from zero. Could you show me?



Any further elaboration on the motivation and purpose of homogeneous coordinates would be appreciated (I know this is the type of vauge/big question that this StackExchange may not like, but I do computer vision, and while I have a vague notion of the motivation and purpose of homogeneous coordinates, another attempt at explaining them to me would never hurt...)










share|cite|improve this question






















  • HINT: What does the cross product of two elements of $mathbb R^3$ look like when expanded in coordinates?
    – amd
    Nov 28 '18 at 21:36
















0














I'm working through an old textbook called Algebraic Projective Geometry, by Semple and Kneebone.



Early in the text, the authors write:




When we introduced complex points (on p. 12) we explained that they are to be regarded as ideal points adjoined to the euclidean plane; and the same is now true of points at infinity. These too are ideal points, but since they have not been introduced by way of coordinate representation it is not immediately obvious how they can be handled algebraically. This can, in fact, be managed quite easily, as we shall now show, by making the coordinates homogeneous.



When a pair of axes has been chosen, all actual (i.e. non-ideal points of the plane admit of unique and unexceptional representation by pairs of real numbers (X, Y). Two lines, whose equations are



aX + bY + c = 0 and a'X + b'Y + c' = 0,



meet, if they are not parallel, in the actual point whose coordinates are given by



X : Y : 1 = bc' - b'c : ca' - c'a : ab' - a'b.



Now whether or not the lines are parallel, supposing only that they are distinct, the ratios



bc' - b'c : ca' - c'a : ab' - a'b



are always determinate, in the sense that at least one of the quantities concerned is different from zero. Our object, then, must be to represent points by such sets of ratios rather than by pairs of numbers which are liable to become infinite.




I don't understand how the ratios are always determinate. I can't see that at least one of the quantities concerned is different from zero. Could you show me?



Any further elaboration on the motivation and purpose of homogeneous coordinates would be appreciated (I know this is the type of vauge/big question that this StackExchange may not like, but I do computer vision, and while I have a vague notion of the motivation and purpose of homogeneous coordinates, another attempt at explaining them to me would never hurt...)










share|cite|improve this question






















  • HINT: What does the cross product of two elements of $mathbb R^3$ look like when expanded in coordinates?
    – amd
    Nov 28 '18 at 21:36














0












0








0







I'm working through an old textbook called Algebraic Projective Geometry, by Semple and Kneebone.



Early in the text, the authors write:




When we introduced complex points (on p. 12) we explained that they are to be regarded as ideal points adjoined to the euclidean plane; and the same is now true of points at infinity. These too are ideal points, but since they have not been introduced by way of coordinate representation it is not immediately obvious how they can be handled algebraically. This can, in fact, be managed quite easily, as we shall now show, by making the coordinates homogeneous.



When a pair of axes has been chosen, all actual (i.e. non-ideal points of the plane admit of unique and unexceptional representation by pairs of real numbers (X, Y). Two lines, whose equations are



aX + bY + c = 0 and a'X + b'Y + c' = 0,



meet, if they are not parallel, in the actual point whose coordinates are given by



X : Y : 1 = bc' - b'c : ca' - c'a : ab' - a'b.



Now whether or not the lines are parallel, supposing only that they are distinct, the ratios



bc' - b'c : ca' - c'a : ab' - a'b



are always determinate, in the sense that at least one of the quantities concerned is different from zero. Our object, then, must be to represent points by such sets of ratios rather than by pairs of numbers which are liable to become infinite.




I don't understand how the ratios are always determinate. I can't see that at least one of the quantities concerned is different from zero. Could you show me?



Any further elaboration on the motivation and purpose of homogeneous coordinates would be appreciated (I know this is the type of vauge/big question that this StackExchange may not like, but I do computer vision, and while I have a vague notion of the motivation and purpose of homogeneous coordinates, another attempt at explaining them to me would never hurt...)










share|cite|improve this question













I'm working through an old textbook called Algebraic Projective Geometry, by Semple and Kneebone.



Early in the text, the authors write:




When we introduced complex points (on p. 12) we explained that they are to be regarded as ideal points adjoined to the euclidean plane; and the same is now true of points at infinity. These too are ideal points, but since they have not been introduced by way of coordinate representation it is not immediately obvious how they can be handled algebraically. This can, in fact, be managed quite easily, as we shall now show, by making the coordinates homogeneous.



When a pair of axes has been chosen, all actual (i.e. non-ideal points of the plane admit of unique and unexceptional representation by pairs of real numbers (X, Y). Two lines, whose equations are



aX + bY + c = 0 and a'X + b'Y + c' = 0,



meet, if they are not parallel, in the actual point whose coordinates are given by



X : Y : 1 = bc' - b'c : ca' - c'a : ab' - a'b.



Now whether or not the lines are parallel, supposing only that they are distinct, the ratios



bc' - b'c : ca' - c'a : ab' - a'b



are always determinate, in the sense that at least one of the quantities concerned is different from zero. Our object, then, must be to represent points by such sets of ratios rather than by pairs of numbers which are liable to become infinite.




I don't understand how the ratios are always determinate. I can't see that at least one of the quantities concerned is different from zero. Could you show me?



Any further elaboration on the motivation and purpose of homogeneous coordinates would be appreciated (I know this is the type of vauge/big question that this StackExchange may not like, but I do computer vision, and while I have a vague notion of the motivation and purpose of homogeneous coordinates, another attempt at explaining them to me would never hurt...)







projective-geometry






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 28 '18 at 19:40









mannyglover

11




11












  • HINT: What does the cross product of two elements of $mathbb R^3$ look like when expanded in coordinates?
    – amd
    Nov 28 '18 at 21:36


















  • HINT: What does the cross product of two elements of $mathbb R^3$ look like when expanded in coordinates?
    – amd
    Nov 28 '18 at 21:36
















HINT: What does the cross product of two elements of $mathbb R^3$ look like when expanded in coordinates?
– amd
Nov 28 '18 at 21:36




HINT: What does the cross product of two elements of $mathbb R^3$ look like when expanded in coordinates?
– amd
Nov 28 '18 at 21:36










2 Answers
2






active

oldest

votes


















0














Think of this:



$$bc'-b'c=0iff bc'=b'cimpliedby b:b'=c:c'$$



So essentially a zero entry in the equation for the point of intersection means that the ratio between the coefficients for one and that for the other line are equal. I said “essentially” because there might be zeros in there, making the rightmost implication not quite an equivalence. I'll cover that in a moment, after explaining what I'm aiming for.



Assume that all three terms in the result are zero. Then you get three such equations, which taken together express $a:a'=b:b'=c:c'$. So the ratio between the coefficients is the same, which means one line equation is just a multiple of the other. If the equations are multiples of one another, they describe the same line. This is excluded by assuming distinct lines.



But what if one of the denominators at the right is zero? Then you can't divide by it in the normal sense. Assume without loss of generality that $b'$ is zero. Then the middle equation becomes $bc'=0$ so either $b$ or $c'$ have to be zero as well. And since you have a similar equation $ab'=a'b$ from another zero in the result, you also need $b$ or $a'$ to be zero. If you take $b=0$, the above argument still holds: one equation is a multiple of the other, even if one of the coefficients happens to be zero. If instead you assume $bneq 0$, then you have to take $a'=0$ and $c'=0$, summarized as $a'=b'=c'=0$, and the second equation becomes null, not representing a line any more.



So if the two inputs are actually lines (i.e. have at least one non-zero coefficient, which will describe the line at infinity for $a=b=0, cneq0$), and if the lines are distinct (one line equation is not a multiple of the other), then the point of intersection computed as described will have at least one non-zero coordinate as well.






share|cite|improve this answer





























    0














    Viewing the homogeneous coordinates of a point as elements of $mathbb R^3$, we can write $mathbf x = (X,Y,1)^T$ and $mathbf l = (a,b,c)$ so that the equation $aX+bY+c=0$ of a line becomes $mathbf l^Tmathbf x = 0$. This is just the dot product of the two vectors, and in $mathbb R^3$ is the equation of a plane through the origin with normal $mathbf l$. If two such planes intersect in a line, then this line is orthogonal to both normals, so its direction vector is the cross product of the two normals: $$mathbf ltimesmathbf m = (bc'-b'c,ca'-c'a,ab'-a'b)^T.$$ The two normals are nonzero, so the only way for this cross product to vanish is for $mathbf l$ and $mathbf m$ to be parallel, but in that case they represent the same plane—in the projective plane, the two lines are not distinct.



    Now, in this model of the projective plane, we’re really working with equivalence classes of points in $mathbb R^3setminus {mathbf 0}$: The homogeneous coordinate vectors $(X,Y,Z)$ and $(kX,kY,kZ)$ correspond to the same point in $mathbb P^2$ and it’s the ratios $X:Y:Z$ that serve to distinguish distinct points in $mathbb P^2$. The intersection of the two lines $mathbf l^Tmathbf x=0$ and $mathbf m^Tmathbf x=0$ is, per the above, then, has homogeneous coordinates with ratios $bc'-b'c:ca'-c'a:ab'-a'b$, which are not all zero if the lines are distinct.






    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',
      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%2f3017590%2fmotivation-for-2d-homogeneous-coordinates-ratio-always-determinate%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      0














      Think of this:



      $$bc'-b'c=0iff bc'=b'cimpliedby b:b'=c:c'$$



      So essentially a zero entry in the equation for the point of intersection means that the ratio between the coefficients for one and that for the other line are equal. I said “essentially” because there might be zeros in there, making the rightmost implication not quite an equivalence. I'll cover that in a moment, after explaining what I'm aiming for.



      Assume that all three terms in the result are zero. Then you get three such equations, which taken together express $a:a'=b:b'=c:c'$. So the ratio between the coefficients is the same, which means one line equation is just a multiple of the other. If the equations are multiples of one another, they describe the same line. This is excluded by assuming distinct lines.



      But what if one of the denominators at the right is zero? Then you can't divide by it in the normal sense. Assume without loss of generality that $b'$ is zero. Then the middle equation becomes $bc'=0$ so either $b$ or $c'$ have to be zero as well. And since you have a similar equation $ab'=a'b$ from another zero in the result, you also need $b$ or $a'$ to be zero. If you take $b=0$, the above argument still holds: one equation is a multiple of the other, even if one of the coefficients happens to be zero. If instead you assume $bneq 0$, then you have to take $a'=0$ and $c'=0$, summarized as $a'=b'=c'=0$, and the second equation becomes null, not representing a line any more.



      So if the two inputs are actually lines (i.e. have at least one non-zero coefficient, which will describe the line at infinity for $a=b=0, cneq0$), and if the lines are distinct (one line equation is not a multiple of the other), then the point of intersection computed as described will have at least one non-zero coordinate as well.






      share|cite|improve this answer


























        0














        Think of this:



        $$bc'-b'c=0iff bc'=b'cimpliedby b:b'=c:c'$$



        So essentially a zero entry in the equation for the point of intersection means that the ratio between the coefficients for one and that for the other line are equal. I said “essentially” because there might be zeros in there, making the rightmost implication not quite an equivalence. I'll cover that in a moment, after explaining what I'm aiming for.



        Assume that all three terms in the result are zero. Then you get three such equations, which taken together express $a:a'=b:b'=c:c'$. So the ratio between the coefficients is the same, which means one line equation is just a multiple of the other. If the equations are multiples of one another, they describe the same line. This is excluded by assuming distinct lines.



        But what if one of the denominators at the right is zero? Then you can't divide by it in the normal sense. Assume without loss of generality that $b'$ is zero. Then the middle equation becomes $bc'=0$ so either $b$ or $c'$ have to be zero as well. And since you have a similar equation $ab'=a'b$ from another zero in the result, you also need $b$ or $a'$ to be zero. If you take $b=0$, the above argument still holds: one equation is a multiple of the other, even if one of the coefficients happens to be zero. If instead you assume $bneq 0$, then you have to take $a'=0$ and $c'=0$, summarized as $a'=b'=c'=0$, and the second equation becomes null, not representing a line any more.



        So if the two inputs are actually lines (i.e. have at least one non-zero coefficient, which will describe the line at infinity for $a=b=0, cneq0$), and if the lines are distinct (one line equation is not a multiple of the other), then the point of intersection computed as described will have at least one non-zero coordinate as well.






        share|cite|improve this answer
























          0












          0








          0






          Think of this:



          $$bc'-b'c=0iff bc'=b'cimpliedby b:b'=c:c'$$



          So essentially a zero entry in the equation for the point of intersection means that the ratio between the coefficients for one and that for the other line are equal. I said “essentially” because there might be zeros in there, making the rightmost implication not quite an equivalence. I'll cover that in a moment, after explaining what I'm aiming for.



          Assume that all three terms in the result are zero. Then you get three such equations, which taken together express $a:a'=b:b'=c:c'$. So the ratio between the coefficients is the same, which means one line equation is just a multiple of the other. If the equations are multiples of one another, they describe the same line. This is excluded by assuming distinct lines.



          But what if one of the denominators at the right is zero? Then you can't divide by it in the normal sense. Assume without loss of generality that $b'$ is zero. Then the middle equation becomes $bc'=0$ so either $b$ or $c'$ have to be zero as well. And since you have a similar equation $ab'=a'b$ from another zero in the result, you also need $b$ or $a'$ to be zero. If you take $b=0$, the above argument still holds: one equation is a multiple of the other, even if one of the coefficients happens to be zero. If instead you assume $bneq 0$, then you have to take $a'=0$ and $c'=0$, summarized as $a'=b'=c'=0$, and the second equation becomes null, not representing a line any more.



          So if the two inputs are actually lines (i.e. have at least one non-zero coefficient, which will describe the line at infinity for $a=b=0, cneq0$), and if the lines are distinct (one line equation is not a multiple of the other), then the point of intersection computed as described will have at least one non-zero coordinate as well.






          share|cite|improve this answer












          Think of this:



          $$bc'-b'c=0iff bc'=b'cimpliedby b:b'=c:c'$$



          So essentially a zero entry in the equation for the point of intersection means that the ratio between the coefficients for one and that for the other line are equal. I said “essentially” because there might be zeros in there, making the rightmost implication not quite an equivalence. I'll cover that in a moment, after explaining what I'm aiming for.



          Assume that all three terms in the result are zero. Then you get three such equations, which taken together express $a:a'=b:b'=c:c'$. So the ratio between the coefficients is the same, which means one line equation is just a multiple of the other. If the equations are multiples of one another, they describe the same line. This is excluded by assuming distinct lines.



          But what if one of the denominators at the right is zero? Then you can't divide by it in the normal sense. Assume without loss of generality that $b'$ is zero. Then the middle equation becomes $bc'=0$ so either $b$ or $c'$ have to be zero as well. And since you have a similar equation $ab'=a'b$ from another zero in the result, you also need $b$ or $a'$ to be zero. If you take $b=0$, the above argument still holds: one equation is a multiple of the other, even if one of the coefficients happens to be zero. If instead you assume $bneq 0$, then you have to take $a'=0$ and $c'=0$, summarized as $a'=b'=c'=0$, and the second equation becomes null, not representing a line any more.



          So if the two inputs are actually lines (i.e. have at least one non-zero coefficient, which will describe the line at infinity for $a=b=0, cneq0$), and if the lines are distinct (one line equation is not a multiple of the other), then the point of intersection computed as described will have at least one non-zero coordinate as well.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 28 '18 at 22:51









          MvG

          30.7k449101




          30.7k449101























              0














              Viewing the homogeneous coordinates of a point as elements of $mathbb R^3$, we can write $mathbf x = (X,Y,1)^T$ and $mathbf l = (a,b,c)$ so that the equation $aX+bY+c=0$ of a line becomes $mathbf l^Tmathbf x = 0$. This is just the dot product of the two vectors, and in $mathbb R^3$ is the equation of a plane through the origin with normal $mathbf l$. If two such planes intersect in a line, then this line is orthogonal to both normals, so its direction vector is the cross product of the two normals: $$mathbf ltimesmathbf m = (bc'-b'c,ca'-c'a,ab'-a'b)^T.$$ The two normals are nonzero, so the only way for this cross product to vanish is for $mathbf l$ and $mathbf m$ to be parallel, but in that case they represent the same plane—in the projective plane, the two lines are not distinct.



              Now, in this model of the projective plane, we’re really working with equivalence classes of points in $mathbb R^3setminus {mathbf 0}$: The homogeneous coordinate vectors $(X,Y,Z)$ and $(kX,kY,kZ)$ correspond to the same point in $mathbb P^2$ and it’s the ratios $X:Y:Z$ that serve to distinguish distinct points in $mathbb P^2$. The intersection of the two lines $mathbf l^Tmathbf x=0$ and $mathbf m^Tmathbf x=0$ is, per the above, then, has homogeneous coordinates with ratios $bc'-b'c:ca'-c'a:ab'-a'b$, which are not all zero if the lines are distinct.






              share|cite|improve this answer


























                0














                Viewing the homogeneous coordinates of a point as elements of $mathbb R^3$, we can write $mathbf x = (X,Y,1)^T$ and $mathbf l = (a,b,c)$ so that the equation $aX+bY+c=0$ of a line becomes $mathbf l^Tmathbf x = 0$. This is just the dot product of the two vectors, and in $mathbb R^3$ is the equation of a plane through the origin with normal $mathbf l$. If two such planes intersect in a line, then this line is orthogonal to both normals, so its direction vector is the cross product of the two normals: $$mathbf ltimesmathbf m = (bc'-b'c,ca'-c'a,ab'-a'b)^T.$$ The two normals are nonzero, so the only way for this cross product to vanish is for $mathbf l$ and $mathbf m$ to be parallel, but in that case they represent the same plane—in the projective plane, the two lines are not distinct.



                Now, in this model of the projective plane, we’re really working with equivalence classes of points in $mathbb R^3setminus {mathbf 0}$: The homogeneous coordinate vectors $(X,Y,Z)$ and $(kX,kY,kZ)$ correspond to the same point in $mathbb P^2$ and it’s the ratios $X:Y:Z$ that serve to distinguish distinct points in $mathbb P^2$. The intersection of the two lines $mathbf l^Tmathbf x=0$ and $mathbf m^Tmathbf x=0$ is, per the above, then, has homogeneous coordinates with ratios $bc'-b'c:ca'-c'a:ab'-a'b$, which are not all zero if the lines are distinct.






                share|cite|improve this answer
























                  0












                  0








                  0






                  Viewing the homogeneous coordinates of a point as elements of $mathbb R^3$, we can write $mathbf x = (X,Y,1)^T$ and $mathbf l = (a,b,c)$ so that the equation $aX+bY+c=0$ of a line becomes $mathbf l^Tmathbf x = 0$. This is just the dot product of the two vectors, and in $mathbb R^3$ is the equation of a plane through the origin with normal $mathbf l$. If two such planes intersect in a line, then this line is orthogonal to both normals, so its direction vector is the cross product of the two normals: $$mathbf ltimesmathbf m = (bc'-b'c,ca'-c'a,ab'-a'b)^T.$$ The two normals are nonzero, so the only way for this cross product to vanish is for $mathbf l$ and $mathbf m$ to be parallel, but in that case they represent the same plane—in the projective plane, the two lines are not distinct.



                  Now, in this model of the projective plane, we’re really working with equivalence classes of points in $mathbb R^3setminus {mathbf 0}$: The homogeneous coordinate vectors $(X,Y,Z)$ and $(kX,kY,kZ)$ correspond to the same point in $mathbb P^2$ and it’s the ratios $X:Y:Z$ that serve to distinguish distinct points in $mathbb P^2$. The intersection of the two lines $mathbf l^Tmathbf x=0$ and $mathbf m^Tmathbf x=0$ is, per the above, then, has homogeneous coordinates with ratios $bc'-b'c:ca'-c'a:ab'-a'b$, which are not all zero if the lines are distinct.






                  share|cite|improve this answer












                  Viewing the homogeneous coordinates of a point as elements of $mathbb R^3$, we can write $mathbf x = (X,Y,1)^T$ and $mathbf l = (a,b,c)$ so that the equation $aX+bY+c=0$ of a line becomes $mathbf l^Tmathbf x = 0$. This is just the dot product of the two vectors, and in $mathbb R^3$ is the equation of a plane through the origin with normal $mathbf l$. If two such planes intersect in a line, then this line is orthogonal to both normals, so its direction vector is the cross product of the two normals: $$mathbf ltimesmathbf m = (bc'-b'c,ca'-c'a,ab'-a'b)^T.$$ The two normals are nonzero, so the only way for this cross product to vanish is for $mathbf l$ and $mathbf m$ to be parallel, but in that case they represent the same plane—in the projective plane, the two lines are not distinct.



                  Now, in this model of the projective plane, we’re really working with equivalence classes of points in $mathbb R^3setminus {mathbf 0}$: The homogeneous coordinate vectors $(X,Y,Z)$ and $(kX,kY,kZ)$ correspond to the same point in $mathbb P^2$ and it’s the ratios $X:Y:Z$ that serve to distinguish distinct points in $mathbb P^2$. The intersection of the two lines $mathbf l^Tmathbf x=0$ and $mathbf m^Tmathbf x=0$ is, per the above, then, has homogeneous coordinates with ratios $bc'-b'c:ca'-c'a:ab'-a'b$, which are not all zero if the lines are distinct.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 29 '18 at 1:49









                  amd

                  29.2k21050




                  29.2k21050






























                      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%2f3017590%2fmotivation-for-2d-homogeneous-coordinates-ratio-always-determinate%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

                      Probability when a professor distributes a quiz and homework assignment to a class of n students.

                      Aardman Animations

                      Are they similar matrix