# How to obtain an equation of a polygon Classic List Threaded 17 messages Open this post in threaded view
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## How to obtain an equation of a polygon

 Gary Tupper wrote (in part): http://mathforum.org/kb/message.jspa?messageID=9558358> Well, if you have used the software to graph say (y-x+3)(x-y^2)=0 > or (y-|x|)(y+3)=0 & some similar such {exprA * exprB * exprC}=0 etc., > then it would not be unreasonable to ask the student on a test to > provide the equation of the '+' coordinate axes. Note: I'm starting a new thread because what I'm posting might be of sufficient interest to others (lurking now or those stumbling on this at some future time) that I'd rather not burry it in the thread "Software We've Liked" that Gary Tupper's post is in. This reminds me of something neat I came across a few years ago. This was after I left teaching, so I never tried it in a classroom, but it seems to me that it could be used to make a worksheet for a possibly engaging one-day project for precalculus students to work on in groups. The method below shows how you can obtain an explicit equation whose graph is a specified polygon in the coordinate plane. I'll begin by making a couple of observations on how one can build new graphs from existing graphs. Let F and G be the graphs (considered as subsets of the coordinate plane) of f(x,y) = 0 and g(x,y) = 0. Then, using * for multiplication, we have the following two useful ways of obtaining other sets as graphs of an equation: "F union G" is the graph of f(x,y) * g(x,y) = 0 "F intersect G" is the graph of |f(x,y)| + |g(x,y)| = 0 For intersection you can also use the graph of [f(x,y)]^2 + [g(x,y)]^2 = 0 if you want to avoid using the absolute value function (e.g., if you wanted the end result to be the zero-set of a 2-variable polynomial). The basic idea is to determine how to obtain an equation whose graph is any specified line segment, and then use the first building principle above to obtain an equation whose graph is any specified set that can be expressed as a finite union of line segments. I'll show how to obtain an equation for the line segment with endpoints (1,0) and (0,1) by a method that can be easily adapted to obtain an equation for a line segment with any specified endpoints (a,b) and (c,d). First, consider the line passing through (1,0) and (0,1). Write this in the form of f(x,y) = 0. One possibility is x + y - 1 = 0. Consider what additional conditions need to be imposed so that we get the line segment with endpoints (1,0) and (0,1), rather than the whole line. In the case of x + y - 1 = 0, we want x >= 0 and x <= 1. Now let's express these inequality conditions as equations. One way to do this is |x| - x = 0 (for x >= 0) and |x - 1| + (x - 1) = 0 (for x <= 1). This comes directly from the definition of absolute value. Recall |u| = u iff u >= 0, and |u| = -u iff u <= 0. The line segment with endpoints (1,0) and (0,1) arises as the intersection of these three sets -- the graph of x + y - 1 = 0, the graph of |x| - x = 0 (gives a half-plane), the graph of |x - 1| + (x - 1) = 0 (gives another half-plane). Therefore, this intersection (i.e. the desired line segment) can be obtained as the graph of |x + y - 1| + ||x| - x| + ||x - 1| + (x - 1)| = 0 A similar method can be used to obtain the line segment with endpoints (a,b) and (c,d) as the graph of an equation of the form f(x,y) = 0. For polygons (or anything that can be expressed as a union of finitely many line segments in the plane), you just obtain equations f_1(x,y) = 0, f_2(x,y) = 0, ..., f_n(x,y) = 0 for each of the line segments, and then the desired planar set will be the graph of f_1(x,y) * f_2(x,y) * ... * f_n(x,y) = 0. Dave L. Renfro
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## Re: How to obtain an equation of a polygon

 Wouldn’t it be a better exercise for the students to derive a general form for this or to prove (show) that the union and intersection constructions work. PS: Just off the top of my head, it seems that the union of plane intersections would form a solid polygon of the same shape. Bob Hansen On Aug 19, 2014, at 4:13 PM, Dave L. Renfro <[hidden email]> wrote: > Gary Tupper wrote (in part): > > http://mathforum.org/kb/message.jspa?messageID=9558358> >> Well, if you have used the software to graph say (y-x+3)(x-y^2)=0 >> or (y-|x|)(y+3)=0 & some similar such {exprA * exprB * exprC}=0 etc., >> then it would not be unreasonable to ask the student on a test to >> provide the equation of the '+' coordinate axes. > > Note: I'm starting a new thread because what I'm posting might be > of sufficient interest to others (lurking now or those stumbling > on this at some future time) that I'd rather not burry it in the > thread "Software We've Liked" that Gary Tupper's post is in. > > This reminds me of something neat I came across a few years ago. > This was after I left teaching, so I never tried it in a classroom, > but it seems to me that it could be used to make a worksheet for a > possibly engaging one-day project for precalculus students to work > on in groups. The method below shows how you can obtain an explicit > equation whose graph is a specified polygon in the coordinate plane. > > I'll begin by making a couple of observations on how one can > build new graphs from existing graphs. > > Let F and G be the graphs (considered as subsets of the coordinate > plane) of f(x,y) = 0 and g(x,y) = 0. Then, using * for multiplication, > we have the following two useful ways of obtaining other sets > as graphs of an equation: > > "F union G" is the graph of f(x,y) * g(x,y) = 0 > > "F intersect G" is the graph of |f(x,y)| + |g(x,y)| = 0 > > For intersection you can also use the graph of > [f(x,y)]^2 + [g(x,y)]^2 = 0 if you want to avoid using > the absolute value function (e.g., if you wanted the > end result to be the zero-set of a 2-variable polynomial). > > The basic idea is to determine how to obtain an equation > whose graph is any specified line segment, and then use > the first building principle above to obtain an equation > whose graph is any specified set that can be expressed as > a finite union of line segments. > > I'll show how to obtain an equation for the line segment > with endpoints (1,0) and (0,1) by a method that can be easily > adapted to obtain an equation for a line segment with > any specified endpoints (a,b) and (c,d). > > First, consider the line passing through (1,0) and (0,1). > > Write this in the form of f(x,y) = 0. One possibility > is x + y - 1 = 0. > > Consider what additional conditions need to be imposed so > that we get the line segment with endpoints (1,0) and (0,1), > rather than the whole line. In the case of x + y - 1 = 0, > we want x >= 0 and x <= 1. > > Now let's express these inequality conditions as equations. > One way to do this is |x| - x = 0 (for x >= 0) > and |x - 1| + (x - 1) = 0 (for x <= 1). > > This comes directly from the definition of absolute value. > Recall |u| = u iff u >= 0, and |u| = -u iff u <= 0. > > The line segment with endpoints (1,0) and (0,1) arises as > the intersection of these three sets -- the graph of > x + y - 1 = 0, the graph of |x| - x = 0 (gives a half-plane), > the graph of |x - 1| + (x - 1) = 0 (gives another half-plane). > Therefore, this intersection (i.e. the desired line segment) > can be obtained as the graph of > > |x + y - 1| + ||x| - x| + ||x - 1| + (x - 1)| = 0 > > A similar method can be used to obtain the line segment > with endpoints (a,b) and (c,d) as the graph of an equation > of the form f(x,y) = 0. > > For polygons (or anything that can be expressed as a union > of finitely many line segments in the plane), you just obtain > equations f_1(x,y) = 0, f_2(x,y) = 0, ..., f_n(x,y) = 0 for > each of the line segments, and then the desired planar set > will be the graph of > > f_1(x,y) * f_2(x,y) * ... * f_n(x,y) = 0. > > Dave L. Renfro
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## Re: How to obtain an equation of a polygon

 On Aug 19, 2014, at 7:06 PM, Robert Hansen <[hidden email]> wrote: > PS: Just off the top of my head, it seems that the union of plane intersections would form a solid polygon of the same shape. I meant to say the intersection of the intersections of the planes would form a solid polygon. The union of the intersections of the planes would form a plane. Bob Hansen
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## Re: How to obtain an equation of a polygon

 In reply to this post by Dave L. Renfro Greetings, Dave: My interest in graphing software derives from my having created a document to display how my son's graphing program could be utilized in the study of elementary algebra. I am of the opinion that the software can provide independent verification of the user's derivation of some particular shape or locus. Some of the notions you alluded to are included in the document. Perhaps of particular interest: the equation of a regular polygon can be expressed concisely using the modulo operator. You can download a copy of the document from: http://www.peda.com/gary/GrafEqApplications.pdfGary Tupper On 19/08/2014 1:13 PM, Dave L. Renfro wrote: > Gary Tupper wrote (in part): > > http://mathforum.org/kb/message.jspa?messageID=9558358> >> Well, if you have used the software to graph say (y-x+3)(x-y^2)=0 >> or (y-|x|)(y+3)=0 & some similar such {exprA * exprB * exprC}=0 etc., >> then it would not be unreasonable to ask the student on a test to >> provide the equation of the '+' coordinate axes. > Note: I'm starting a new thread because what I'm posting might be > of sufficient interest to others (lurking now or those stumbling > on this at some future time) that I'd rather not burry it in the > thread "Software We've Liked" that Gary Tupper's post is in. > > This reminds me of something neat I came across a few years ago. > This was after I left teaching, so I never tried it in a classroom, > but it seems to me that it could be used to make a worksheet for a > possibly engaging one-day project for precalculus students to work > on in groups. The method below shows how you can obtain an explicit > equation whose graph is a specified polygon in the coordinate plane. > > I'll begin by making a couple of observations on how one can > build new graphs from existing graphs. > > Let F and G be the graphs (considered as subsets of the coordinate > plane) of f(x,y) = 0 and g(x,y) = 0. Then, using * for multiplication, > we have the following two useful ways of obtaining other sets > as graphs of an equation: > > "F union G" is the graph of f(x,y) * g(x,y) = 0 > > "F intersect G" is the graph of |f(x,y)| + |g(x,y)| = 0 > > For intersection you can also use the graph of > [f(x,y)]^2 + [g(x,y)]^2 = 0 if you want to avoid using > the absolute value function (e.g., if you wanted the > end result to be the zero-set of a 2-variable polynomial). > > The basic idea is to determine how to obtain an equation > whose graph is any specified line segment, and then use > the first building principle above to obtain an equation > whose graph is any specified set that can be expressed as > a finite union of line segments. > > I'll show how to obtain an equation for the line segment > with endpoints (1,0) and (0,1) by a method that can be easily > adapted to obtain an equation for a line segment with > any specified endpoints (a,b) and (c,d). > > First, consider the line passing through (1,0) and (0,1). > > Write this in the form of f(x,y) = 0. One possibility > is x + y - 1 = 0. > > Consider what additional conditions need to be imposed so > that we get the line segment with endpoints (1,0) and (0,1), > rather than the whole line. In the case of x + y - 1 = 0, > we want x >= 0 and x <= 1. > > Now let's express these inequality conditions as equations. > One way to do this is |x| - x = 0 (for x >= 0) > and |x - 1| + (x - 1) = 0 (for x <= 1). > > This comes directly from the definition of absolute value. > Recall |u| = u iff u >= 0, and |u| = -u iff u <= 0. > > The line segment with endpoints (1,0) and (0,1) arises as > the intersection of these three sets -- the graph of > x + y - 1 = 0, the graph of |x| - x = 0 (gives a half-plane), > the graph of |x - 1| + (x - 1) = 0 (gives another half-plane). > Therefore, this intersection (i.e. the desired line segment) > can be obtained as the graph of > > |x + y - 1| + ||x| - x| + ||x - 1| + (x - 1)| = 0 > > A similar method can be used to obtain the line segment > with endpoints (a,b) and (c,d) as the graph of an equation > of the form f(x,y) = 0. > > For polygons (or anything that can be expressed as a union > of finitely many line segments in the plane), you just obtain > equations f_1(x,y) = 0, f_2(x,y) = 0, ..., f_n(x,y) = 0 for > each of the line segments, and then the desired planar set > will be the graph of > > f_1(x,y) * f_2(x,y) * ... * f_n(x,y) = 0. > > Dave L. Renfro
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## Re: How to obtain an equation of a polygon

 In reply to this post by Dave L. Renfro Referring to Dave Renfro's dt. Aug 20, 2014 1:43 AM (http://mathforum.org/kb/message.jspa?messageID=9567096), on developing an explicit equation for a specified polygon in the coordinate plane: Very neat indeed.  I shall pass it on to the math teachers I meet from time to time.  There are quite a few useful math lessons hidden in there, and perhaps (I should think) some indicators as well of how computers could be useful for enhancing the effectiveness of learning and teaching of math.  I'm sure the keen and imaginative teacher should be able to take off from there in any one of a number of directions. Referring to Robert Hansen's (RH's) responses dt. Aug 20, 2014 4:36 AM (http://mathforum.org/kb/message.jspa?messageID=9567147) and dt Aug 20, 2014 5:02 AM (http://mathforum.org/kb/message.jspa?messageID=9567160):I guess there's no way at all to ENCOURAGE, PUSH or even GOAD Robert Hansen to express himself in simple, clear, 'unsubtle'  language.   For the  life of me, I was unable to make out what Mr Hansen was seeking to convey. (I am now planning to sign up for a course of 'American' English poetry, which hopefully may help me to understand the 'subtleties' that may be concealed in Mr Hansen's remarkable postings.  Only minor 'sarcasm' here).   I believe, however, that it's not so much a matter of concealed subtleties in the messages in question as one of a real inability to write clearly and correctly what is desired to be communicated - or one of plain and simple  intellectual laziness.  From various indicators shown by Mr Hansen over the years, I suspect the latter. Meanwhile, if Mr Hansen or anyone else could throw some light on what it was that Mr Hansen was tryng to tell us, I'm sure it would be very helpful indeed, to me and others here. GSC
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## Re: How to obtain an equation of a polygon

 On Aug 20, 2014, at 3:03 AM, GS Chandy <[hidden email]> wrote: > Meanwhile, if Mr Hansen or anyone else could throw some light on what it was that Mr Hansen was tryng to tell us, I'm sure it would be very helpful indeed, to me and others here. Regarding Dave’s activity… If the activity is essentially turning the knobs on an etch-a-sketch, then I fail to see the math in that. If the activity is really a culmination of awareness in polynomial factoring, absolute value, zero products, the fundamental theorem of algebra, etc. then that is something else. Regarding Gary’s document... I am intrigued that Gary’s document identifies the same union/intersection strategies Dave outlined. I understand the technical aspects of the GraphEq approach to graphing, and I think Gary’s document does a decent job delving into all of that. But as to the pedagogy of his exercises and the gist of what this all purports to teach, it is atrocious. Either Gary’s students are advanced, they understand the math in this document and the exercises are just a diversion. Or If my son came home from school doing these exercises and didn’t understand a lick of what was going on (mathematically), I would be at the school in no time. There is certainly a place for activities. But I question the authenticity when the math behind the activity and the activity itself are at such different levels. Gary has shown us a form of activity (as has Dave). Maybe they can supply some background on the students’ level of knowledge and awareness and what they are trying to accomplish with these activities. I am pretty sure (but I could be wrong) that Dave’s activity is just something neat for a algebra 2 class to do after they have succeeded mostly with the class. Gary’s lessons seem aimed at students that can’t do anything but graph. Bob Hansen
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## Re: How to obtain an equation of a polygon

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## Re: How to obtain an equation of a polygon

 On Aug 20, 2014, at 10:41 AM, GS Chandy <[hidden email]> wrote: > 5.  I believe, offhand, that GraphEq and this manual together should put paid very effectively indeed to Robert Hansen's supercilious (and pretty foolish) question: > > "How is this even pertinent to teaching math?” The document is mathematical, and seems to be geared to teachers. If the document is geared towards students, then why are the exercises geared towards toddlers? It would be as if I wrote a document on the morphology of greek words, with much detail, and then, for exercises, had the students match words to pictures. Or a document on the mathematics behind GPS, with much detail, and then, for the exercise, a scavenger hunt. Bob Hansen
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## Re: How to obtain an equation of a polygon

 In reply to this post by Robert Hansen Robert:The document was conceived as a tool for those wishing to learn mathematics beyond what is currently covered in most current curricula. (in comparison to many enrichment programs that are intended to prepare students for subsequent courses.) At the calculus level, if a student was presented with a particularly difficult derivative exercise, and after many lines of calculation, was able to arrive at a maximum point - but was not confident s/he had the correct answer, then the software might be of value. Note that when a relation is plotted and we wish to determine the maximum, if we zoom into the curve at that location, the curve 'linearizes'. This problem affords the student an opportunity to apply a technique learned in algebra: the 'compression' transformation effected by replacing x with nx to squeeze the curve & thus accentuate peaks. At the trig level, I've attempted to display the link between the circular and hyperbolic , and show how these notions can be extended to other shapes.I tend to think of math as a set of tools that can be applied to a wide range of situations that can be adequately clearly specified.  I speculate that in a typical class of 30 students, there may be one or two who would find the items in the document of interest. Most classmates are likely to ask "Will this be on the test?", and these topics are not really for them. Gary TupperPs There is little to be intrigued about wrt Dave and me using the same strategies: math is math.On Wednesday, August 20, 2014, Robert Hansen <[hidden email]> wrote: On Aug 20, 2014, at 3:03 AM, GS Chandy <chandy.sag@...> wrote: > Meanwhile, if Mr Hansen or anyone else could throw some light on what it was that Mr Hansen was tryng to tell us, I'm sure it would be very helpful indeed, to me and others here. Regarding Dave’s activity… If the activity is essentially turning the knobs on an etch-a-sketch, then I fail to see the math in that. If the activity is really a culmination of awareness in polynomial factoring, absolute value, zero products, the fundamental theorem of algebra, etc. then that is something else. Regarding Gary’s document... I am intrigued that Gary’s document identifies the same union/intersection strategies Dave outlined. I understand the technical aspects of the GraphEq approach to graphing, and I think Gary’s document does a decent job delving into all of that. But as to the pedagogy of his exercises and the gist of what this all purports to teach, it is atrocious. Either Gary’s students are advanced, they understand the math in this document and the exercises are just a diversion. Or If my son came home from school doing these exercises and didn’t understand a lick of what was going on (mathematically), I would be at the school in no time. There is certainly a place for activities. But I question the authenticity when the math behind the activity and the activity itself are at such different levels. Gary has shown us a form of activity (as has Dave). Maybe they can supply some background on the students’ level of knowledge and awareness and what they are trying to accomplish with these activities. I am pretty sure (but I could be wrong) that Dave’s activity is just something neat for a algebra 2 class to do after they have succeeded mostly with the class. Gary’s lessons seem aimed at students that can’t do anything but graph. Bob Hansen
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## Re: How to obtain an equation of a polygon

 In reply to this post by Dave L. Renfro Dave L. Renfro wrote (in part): http://mathforum.org/kb/message.jspa?messageID=9567096> [...] The method below shows how you can obtain an explicit equation > whose graph is a specified polygon in the coordinate plane. [...] For those interested in some of the literature on this topic, the 10 papers listed below are fairly elementary (except maybe ) and should be mostly understandable to good high school precalculus students ("good" being roughly top 10% to 20% in an average level class). My post was based on ideas in  and , each of which was written by a high school student.  M. Amit, M. N. Fried, and P. Satianov, "The equation of a triangle", Mathematics Teacher 94 (2001), 362-364.  Robert F. Jolly, "Equations for semicircles and pings", Mathematics Teacher 60 (1967), 720-722.  Harsh Luthar, "Equation of a line segment", Mathematics Student Journal 21 #1 (1973), 3-4.  Clarence R. Perisho, "Curves with corners", Mathematics Teacher 55 (1962), 326-329.  Clarence R. Perisho, "The use of transformations in deriving equations of common geometric figures", Mathematics Teacher 58 (1965), 386-392.  Joseph F. Santner, "A note on curve fitting", Mathematics Teacher 56 (1963), 218-221.  Joseph F. Santner, "A second note on curve fitting", Mathematics Teacher 56 (1963), 307-310.  Charles P. Seguin, "Equations of polygonal paths", American Mathematical Monthly 69 (1962), 548-549.  John L. Spence, "Equations of some common geometric figures", School Science and Mathematics 58 (1958), 674-676.  Edwin F. Wilde, "Equations of polygons", Mathematics Student Journal 19 #2 (1972), 1-3. Dave L. Renfro
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## Re: How to obtain an equation of a polygon

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## Re: How to obtain an equation of a polygon

 In reply to this post by Dave L. Renfro Remarking on Dave Renfro's post dt. Aug 21, 2014 1:35 AM (http://mathforum.org/kb/message.jspa?messageID=9567793):Would it be possible at all to provide links to those 10 papers?   GSC
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## Re: How to obtain an equation of a polygon

 In reply to this post by gatupper The topic is great, for a student that understands, but the exercises (and your remarks) read as if you feel that even if the student doesn't understand the math behind the activity, they will still leave with something useful. After years of searching I have found no evidence that this is the case. Bob Hansen > On Aug 20, 2014, at 12:15 PM, "Gary Tupper" <[hidden email]> wrote: > > I speculate that in a typical class of 30 students, there may be one or two who would find the items in the document of interest. Most classmates are likely to ask "Will this be on the test?", and these topics are not really for them.
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## Re: How to obtain an equation of a polygon

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## Re: How to obtain an equation of a polygon

 I actually looked for that journal as well and came across an article related to its beginning in 1959. It is kind of sad that a “student journal” like that just disappeared. Bob Hansen On Aug 21, 2014, at 10:48 AM, Dave L. Renfro <[hidden email]> wrote: > The remaining two items,  and , were published in a journal > that I can't find on the internet. I'm rather surprised that some > organization such as NCTM hasn't archived digital copies of this > journal.