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Analytic geometry textbooks usually teach the equation of a straight line and the equations of a circle and conic sections (equation of an ellipse/hyperbola/parabola). None (except a russian one) as far I as know deals with the analytic equation of a triangle and/or the analytic equation of a quadrilateral. For some unknown reason this subject seems to be as rare as hen's teeth in analytic geometry primers. Only in russian I've found a book dealing with such cartesian equations, by V. L. Rvatchev (Геометрические приложения алгебры логики, Киев, 1967).

Does anyone know a textbook with a section on the cartesian equation of a triangle/quadrilateral or a book/monograph about this subject? It could be in english, german, french, spanish or portuguese.

MrDudulex
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    The reason that equations of triangles or other polygons are not common is simple. They're not as useful as the equations of lines or conics. Such equations often includes several absolute values and using them to prove something requieres studying several cases so often gives no benefit over representing the polygon with its vertices instead of an equation. – jjagmath Jul 19 '23 at 17:20
  • ... they are also not functions. – David G. Stork Jul 19 '23 at 18:25
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    See Studying set theory using polynomials? (my answer gives some English references for Rvatchev's work) AND Is there an equation to describe regular polygons? AND these other MSE questions. – Dave L. Renfro Jul 19 '23 at 18:31
  • jjagmath, I know that the more sides a polygon has the more absolute values (and nasty nested ones) grow in its equation. But if you limit yourself to the cases n=3 and n=4 (triangle and quadrilateral) and make a convenient choice of the equations of sides, you can arrive at fairly simple equations of the triangle or quadrilateral with interest properties. A equation of a parallelogram, for instance, in its simplest form is abs(Ax +By + C) + abs(Dx +Ey +F) = G, the polynoms in the abs being particular equations of the diagonals. – MrDudulex Jul 19 '23 at 20:06
  • And this simple equation gives you many informations at once: it tells you if the parallelogramm is a square or a rectangle or a rhombus and it gives you the area of the parallelogram. As a bonus, you have also the inequation of the polygon and then you can evaluate if a point is in its interior or not. – MrDudulex Jul 19 '23 at 20:16

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(repeat of my earlier comment) See Studying set theory using polynomials? (my answer gives some English references for Rvatchev's work) AND Is there an equation to describe regular polygons? AND these other MSE questions.

Getting a bit away from your question, but still likely of interest to many who stumble upon this answer in the future, see this answer. El-Milick's book that I cite there is especially interesting, although rather rare and nowhere on the internet that I know of -- one day I might try to make a .pdf file from my copy for the Internet Archive.

I have seen small sections and exercises in a few books that deal with graphs of polygons, but I haven't kept track of these books and I don't know of any specifics at the moment. {{ (update 6 days later) See [0], which the OP told me was mentioned in [7] }} However, I do know of several expository papers that deal with such graphs, and these papers are listed below. (I have a folder of photocopies of such papers, which took me a rather long time to locate, hence the time gap between my earlier comment and this answer.) Of course, there are probably lots of internet writings on this topic -- the MSE references above, people's blogs, unpublished notes at various people's web pages, etc. -- but I'll leave these to you or someone else to find. Anyway, for this topic you seemed more interested in published literature than internet writings and postings.

[0] Anders Leonard Axel Söderblom (1847–1912), Sur l'Emploi de Valeurs Absolues dans la Géométrie Analytique [On the Use of Absolute Values in Analytic Geometry], Wald. Zachrissons Boktryckeri A.-B. (Göteborg, Sweden), 1906, vi + 174 pages. copy 1 & copy 2

Reviews: JFM 38.0592.01 (in German; here also) and Mathesis Recueil Mathématique (in French)

[1] [author not known], [Solution to 558], Mathematical Gazette 4 #63 (April 1907), pp. 53−54. [See also MG 3 #18, December 1904, p. 116.]

(problem statement) Shew that the equation of the five sides of a regular pentagon, when the origin is at the centre of the inscribed circle and the axis of $x$ is perpendicular to one of the sides, may be expressed in the form $$ 5(x^2 + y^2)^2(a-x) \; - \; 20(x^2 + y^2)(a^3 - x^3) \; + \; 16(a^5 - x^5) \; = \; 0. $$

[2] Miriam Amit, Michael N. Fried, and Pavel Satianov, The equation of a triangle, Mathematics Teacher 94 #5 (May 2001), pp. 362−364.

[3] John E. Freund, Segment – functions, Mathematics Magazine 21 #5 (May−June 1948), pp. 261−264.

[4] Manuel Fernandez Guasti, Analytic geometry of some rectilinear figures, International Journal of Mathematical Education in Science and Technology 23 #6 (1992), pp. 895−901.

[5] Robert F. Jolly, Equations for semicircles and pings, Mathematics Teacher 60 #7 (November 1967), pp. 720−722.

[6] Harsh Luthar, Equation of a line segment, Mathematics Student Journal 21 #1 (October 1973), pp. 3−4.

[7] C. O. Oakley, Equations of polygons, American Mathematical Monthly 42 #8 (October 1935), pp. 476−487.

A very thorough treatment.

[8] C. O. Oakley, Equations of polygonal configurations, American Mathematical Monthly 47 #9 (November 1940), pp. 621−627.

[9] Clarence R. Perisho, Curves with corners, Mathematics Teacher 55 #5 (May 1962), pp. 326−329.

[10] Clarence R. Perisho, The use of transformations in deriving equations of common geometric figures, Mathematics Teacher 58 #5 (May 1965), pp. 386−392.

[11] Stanley Rabinowitz; Russell Euler and Jawad Sadek, [Solutions to Problem 592], The Pentagon 66 #1 (Fall 2006), p. 58.

(problem statement, by Rabinowitz) The points $(0,0),$ $(1,0),$ and $(0,1)$ are the vertices of a square $S.$ Find an equation in $x$ and $y$ whose graph in the $xy$-plane is $S.$

(Rabinowitz solution) Start with the equation $|x| + |y| = 1,$ which is the graph of a diamond. Rotate the axes $45^{\text{o}}$ and perform a scaling to get the equation $|x-y| + |x+y-1| = 1.$

(Euler/Sadek solution) One equation that works is $\max (x,\,y) = \min (x+y,\,1).$

[12] Joseph F. Santner, A note on curve fitting, Mathematics Teacher 56 #4 (April 1963), pp. 218−221.

[13] Joseph F. Santner, A second note on curve fitting, Mathematics Teacher 56 #5 (May 1963), pp. 307−310.

[14] Charles P. Seguin, Equations of polygonal paths, American Mathematical Monthly 69 #6 (June−July 1962), pp. 548−549.

[15] John L. Spence, Equations of some common geometric figures, School Science and Mathematics 58 #9 (December 1958), pp. 674−676.

[16] Edwin F. Wilde, Equations of polygons, Mathematics Student Journal 19 #2 (January 1972), pp. 1−3.

[17] See Chin Woon, A regular polygon equation, Pi Mu Epsilon Journal 9 #9 (Fall 1993), pp. 597−598.

Woon obtains an equation in polar coordinates whose graph is a regular polygon with $n$ sides.

Dave L. Renfro
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