Skip to main content
Log in

A Symbolic Dynamic Geometry System Using the Analytical Geometry Method

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

A symbolic geometry system such as Geometry Expressions can generate symbolic measurements in terms of indeterminate inputs from a geometric figure. It has elements of dynamic geometry system and elements of automated theorem prover. Geometry Expressions is based on the analytical geometry method. We describe the method in the style used by expositions of semi-synthetic theorem provers such as the area method. The analytical geometry method differs in that it considers geometry from a traditional Euclidean/Cartesian perspective. To the extent that theorems are proved, they are only proved for figures sufficiently close to the given figure. This clearly has theoretical disadvantages, however they are balanced by the practical advantage that the geometrical model used is familiar to students and engineers. The method decouples constructions from geometrical measurements, and thus admits a broad variety of measurement types and construction types. An algorithm is presented for automatically deriving simple forms for angle expressions and is shown to be equivalent to a class of traditional proofs. A semi-automated proof system comprises the symbolic geometry system, a CAS and the user. The user’s inclusion in the hybrid system is a key pedagogic advantage. A number of examples are presented to illustrate the breadth of applicability of such a system and the user’s role in proof.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Similar content being viewed by others

References

  1. Botana, F., Hohenwarter, M., Janičić, P., Kovács, Z., Petrović, I., Recio, T., Weitzhofer, S.: Automated theorem proving in GeoGebra: current achievements. J. Autom. Reason. 55(1), 39–59 (2015)

    Article  MathSciNet  Google Scholar 

  2. Buchberger, B., Winkler, F. (eds.): Gröbner Bases and Applications, vol. 251. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  3. Chou, S.C.: Mechanical Geometry Theorem Proving, vol. 41. Springer, Berlin (1988)

    MATH  Google Scholar 

  4. Chou, S.C., Gao, X.S., Zhang, J.: Machine Proofs in Geometry: Automated Production of Readable Proofs for geometry Theorems, vol. 6. World Scientific, Singapore (1994)

    Book  Google Scholar 

  5. Dorrie, H.: 100 Great Problems of Elementary Mathematics: Their History and Solution. Dover, Illinois (2013)

    Google Scholar 

  6. Hohenwarter, M., Jones, K.: Ways of linking geometry and algebra, the case of Geogebra. Proc. Br. Soc. Res. Learn. Math. 27(3), 126–131 (2007)

    Google Scholar 

  7. Jackiw, N.: The Geometer’s Sketchpad (computer software). Key Curriculum Press, Emeryville, CA (1991)

    Google Scholar 

  8. Janičić, P., Narboux, J., Quaresma, P.: The area method. J. Autom. Reason. 48(4), 489–532 (2012)

    Article  MathSciNet  Google Scholar 

  9. Kutzler, B., Stifter, S.: On the application of Buchberger’s algorithm to automated geometry theorem proving. J. Symb. Comput. 2(4), 389–397 (1986)

    Article  MathSciNet  Google Scholar 

  10. Laborde, J.M., Bellemain, F.: Cabri Geometry (computer software). Texas Instruments, Dallas TX (1990)

    Google Scholar 

  11. Magajna, Z.: Overcoming the obstacle of poor knowledge in proving geometry tasks. CEPS J. 3(4), 99 (2013)

    Google Scholar 

  12. Prautzsch, H., Boehm, W., Paluszny, M.: Bézier and B-spline Techniques. Springer, Berlin (2013)

    MATH  Google Scholar 

  13. Richtert-Gebert, J., Kortenkamp, U.H.: The Interactive Geometry Software “Cinderella” (computer software). Springer, Berlin (1999)

    MATH  Google Scholar 

  14. Todd, P.: Geometry expressions: a constraint based interactive symbolic geometry system. In: International Workshop on Automated Deduction in Geometry (pp. 189–202). Springer, Berlin (2006)

  15. Todd, P.: A k-tree generalization that characterizes consistency of dimensioned engineering drawings. SIAM J. Discrete Math. 2(2), 255–261 (1989)

    Article  MathSciNet  Google Scholar 

  16. Todd, P.: Application of Geometry Expressions to Theorems from Machine Proofs in Geometry. Saltire Software Technical Report TR2016-1. Available from https://www.saltire.com/download/TR2016-1.pdf (2016)

  17. Wu, W.T.: Mechanical Theorem Proving in Geometries: Basic Principles. Springer, Berlin (2012)

    Google Scholar 

  18. Zou, Y., Zhang, J.: Automated generation of readable proofs for constructive geometry statements with the mass point method. In: International Workshop on Automated Deduction in Geometry (pp. 221–258). Springer, Berlin (2010)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Philip Todd.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This material is based upon work supported, in part, by the National Science Foundation under Grant IIP-0750028.

Appendix A

Appendix A

We present constructions for an implementation whose geometric elements are points and oriented lines. Oriented lines are represented by the triple (ABC), where the underlying line is defined by the equation \(A\cdot x+B\cdot y+C=0\), and the orientation is defined by the vector (AB). Hence the triple \((-A,-B,-C)\) represents the same line as (ABC) but with the opposite orientation. In each case we show the elimination step for the point’s coordinates \((x_{P},y_{P})\) or the line’s coefficients \((A_{L},B_{L},C_{L})\). The equation of the line with these coefficients is \(A_{L}\cdot x+B_{L}\cdot y+C_{L}=0\).

C1 Point given coordinates P is defined to have coordinates (x,y).

$$\begin{aligned}&x_{P}\rightarrow x\\&\begin{array}{c} y_{P}\rightarrow y\end{array} \end{aligned}$$

C2 Point proportion along a segment Point P is proportion k along the segment \(P_{0}P_{1}\).

$$\begin{aligned}&x_{P}\rightarrow k\cdot x_{1}+(1-k)\cdot x_{0}\\&\begin{array}{c} y_{P}\rightarrow k\cdot y_{1}+(1-k)\cdot y_{0}\end{array} \end{aligned}$$

C3 Point distance from two points If point P is distance \(d_{0}\) from point \(P_{0}\) and distance \(d_{1}\) from point \(P_{1}\) and to the left of ordered segment \(P_{0}P_{1}\):

$$\begin{aligned}&x_{P}\rightarrow x_{1}+\dfrac{d_{3}\left( x_{0}-x_{1}\right) }{2d_{2}^{2}} +\dfrac{d_{4}\left( y_{0}-y_{1}\right) }{2d_{2}^{2}}\\&y_{P}\rightarrow y_{1} -\dfrac{d_{4}\left( x_{0}-x_{1}\right) }{2d_{2}^{2}}+\dfrac{d_{3} \left( y_{0}-y_{1}\right) }{2d_{2}^{2}} \end{aligned}$$

If point P is distance \(d_{0}\) from point \(P_{0}\) and distance \(d_{1}\) from point \(P_{1}\) and to the right of ordered segment \(P_{0}P_{1}\):

$$\begin{aligned}&x_{P}\rightarrow x_{0}-\dfrac{d_{5}\left( x_{0}-x_{1}\right) }{2d_{2}^{2}}-\dfrac{d_{4}\left( y_{0}-y_{1}\right) }{2d_{2}^{2}}\\&\begin{array}{c} y_{P}\rightarrow y_{0}+\dfrac{d_{4}\left( x_{0}-x_{1}\right) }{2d_{2}^{2}}-\dfrac{d_{5}\left( y_{0}-y_{1}\right) }{2d_{2}^{2}}\end{array} \end{aligned}$$

where:

$$\begin{aligned} d_{2}= & {} \sqrt{\left( x_{1}-x_{0}\right) ^{2}+\left( y_{1}-y_{0}\right) ^{2}}\\ d_{3}= & {} -d_{0}^{2}+d_{1}^{2}+d_{2}^{2}\\ d_{4}= & {} \sqrt{d_{0}+d_{1}+d_{2}}\sqrt{d_{0}+d_{1}-d_{2}}\sqrt{d_{0}-d_{1}+d_{2}}\sqrt{-d_{0}+d_{1}+d_{2}}\\ d_{5}= & {} d_{0}^{2}-d_{1}^{2}+d_{2}^{2} \end{aligned}$$

C4 Point distance from two lines Point P is directed distance \(d_{0}\) from line \(L_{0}\) and directed distance \(d_{1}\) from line \(L_{1}\).

$$\begin{aligned}&x_{P}\rightarrow \dfrac{-B_{1}\left( C_{0}-d_{0} \sqrt{A_{0}^{2}+B_{0}^{2}}\right) +B_{0}\left( C_{1}-d_{1}\sqrt{A_{1}^{2}+B_{1}^{2}}\right) }{A_{0}B_{1}-A_{1}B_{0}}\\&\begin{array}{c} y_{P}\rightarrow \dfrac{A_{1}\left( C_{0}-d_{0}\sqrt{A_{0}^{2}+B_{0}^{2}}\right) -A_{0}\left( C_{1}-d_{1}\sqrt{A_{1}^{2}+B_{1}^{2}}\right) }{A_{0}B_{1}-A_{1}B_{0}}\end{array} \end{aligned}$$

C5 Point distance from one line and one point We define the 2D wedge product \((x_{0},y_{0})\wedge (x_{1},y_{1})=x_{0}y_{1}-x_{1}y_{0}\)

If point P is directed distance \(d_{0}\) from line \(L_{0}\) and directed distance \(d_{1}\) from point \(P_{1}\) and s \(\overrightarrow{P_{0}P}\wedge (A_{0},B_{0})>0\)

$$\begin{aligned}&x_{P}\rightarrow x_{0}+B_{0}\cdot d_{3}-\dfrac{A_{0}\cdot \left( d_{2}-d_{0}\right) }{\sqrt{A_{0}^{2}+B_{0}^{2}}}\\&\begin{array}{c} y_{P}\rightarrow y_{0}-A_{0}\cdot d_{3}-\dfrac{B_{0}\cdot \left( d_{2}-d_{0}\right) }{\sqrt{A_{0}^{2}+B_{0}^{2}}}\end{array} \end{aligned}$$

If point P is directed distance \(d_{0}\) from line \(L_{0}\) and directed distance \(d_{1}\) from point \(P_{1}\) and \(\overrightarrow{P_{0}P}\wedge (A_{0},B_{0})>0\)

$$\begin{aligned}&x_{P}\rightarrow x_{0}-B_{0}\cdot d_{3}-\dfrac{A_{0}\cdot \left( d_{2}-d_{0}\right) }{\sqrt{A_{0}^{2}+B_{0}^{2}}}\\&\begin{array}{c} y_{P}\rightarrow y_{0}+A_{0}\cdot d_{3}-\dfrac{B_{0}\cdot \left( d_{2}-d_{0}\right) }{\sqrt{A_{0}^{2}+B_{0}^{2}}}\end{array} \end{aligned}$$

where:

$$\begin{aligned}&d_{2}=\dfrac{C_{0}+A_{0}\cdot x_{0}+B_{0}\cdot y_{0}}{\sqrt{A_{0}^{2}+B_{0}^{2}}}\\&\begin{array}{l} \begin{array}{l} d_{3}=\dfrac{\sqrt{d_{1}^{2}-\left( d_{2}-d_{0}\right) ^{2}}}{\sqrt{A_{0}^{2}+B_{0}^{2}}}\end{array}\end{array} \end{aligned}$$

C6 Line given equation If line L is defined by equation \(A\cdot x+B\cdot y+C=0\) and its orientation is aligned with (A,B)

$$\begin{aligned} \begin{array}{c} A_{L}\rightarrow A\\ B_{L}\rightarrow B\\ C_{L}\rightarrow C \end{array} \end{aligned}$$

If line L is defined by equation \(A\cdot x+B\cdot y+C=0\) and its orientation is not aligned with (A,B)

$$\begin{aligned} \begin{array}{c} A_{L}\rightarrow -A\\ B_{L}\rightarrow -B\\ C_{L}\rightarrow -C \end{array} \end{aligned}$$

C7 Line distance from two points Line L is directed distance \(d_{0}\) from point \(P_{0}\) and directed distance \(d_{1}\) from point \(P_{1}\)

$$\begin{aligned}&A_{L}\rightarrow \phi _{0}\\&B_{L}\rightarrow \phi _{1}\\&C_{L}\rightarrow \phi _{2} \end{aligned}$$

where:

$$\begin{aligned} d_{2}= & {} \left( x_{1}-x_{0}\right) ^{2}+\left( y_{1}-y_{0}\right) ^{2}\\ \phi _{0}= & {} \dfrac{\left( d_{1}-d_{0}\right) \left( x_{1}-x_{0}\right) }{d_{2}}+\dfrac{\left( y_{0}-y_{1}\right) \sqrt{d_{2}-\left( d_{0}-d_{1}\right) ^{2}}}{d_{2}}\\ \phi _{1}= & {} \dfrac{\left( d_{0}-d_{1}\right) \left( y_{0}-y_{1}\right) }{d_{2}}+\dfrac{\left( x_{1}-x_{0}\right) \sqrt{d_{2}-\left( d_{0}-d_{1}\right) ^{2}}}{d_{2}}\\ \phi _{2}= & {} d_{1}-x_{1}\cdot \phi _{0}-y_{1}\cdot \phi _{1} \end{aligned}$$

C8 Line angle to a line distance from a point Line L is angle \(\theta _{0}\) from line \(L_{0}\) and directed distance \(d_{1}\) from point \(P_{1}\)

$$\begin{aligned}&A_{L}\rightarrow A_{0}cos\theta _{0}-B_{0}sin\theta _{0}\\&B_{L}\rightarrow A_{0}sin\theta _{0}+B_{0}cos\theta _{0}\\&\begin{array}{c} C_{L}\rightarrow d_{1}\sqrt{A_{0}^{2}+B_{0}^{2}}+B_{0}\left( x_{1}sin\theta _{0}-y_{1}cos\theta _{0}\right) -A_{0}\left( x_{1}cos\theta _{0}+y_{1}\cdot sin\theta _{0}\right) \end{array} \end{aligned}$$

We note that the some of the above constructions have variants dependent on an assessment of sidedness. It is the responsibility of the agent which creates the construction sequence, the dynamic geometry UI for example, to select which variant to insert into the construction sequence. Each of the constructions above fully defines a geometrical object. In a dynamic geometry system, it is common to have geometry which is unconstrained, or partially constrained. For example a point can be located in an arbitrary location, or a point can be placed on a line, or a point can be specified to be at a certain distance from a given point. In such cases, the system can add indeterminates in the process of making the construction. These partial constructions are specified below.

C9 Point distance from one point Point P is distance \(d_{0}\) from point \(P_{0}\)

$$\begin{aligned} \begin{array}{c} x_{P}\rightarrow x_{0}+d_{0}cos\left( \theta _{0}\right) \\ y_{P}\rightarrow y_{0}+d_{0}sin\left( \theta _{0}\right) \end{array} \end{aligned}$$

where \(\theta _{0}\) is an indeterminate introduced by the construction.

C10 Point distance from one line Point P is directed distance \(d_{0}\) from line \(L_{0}\)

$$\begin{aligned} \begin{array}{c} x_{P}\rightarrow \dfrac{B_{0}d_{1}-A_{0}\left( C_{0}-d_{0}\sqrt{A_{0}^{2}+B_{0}^{2}}\right) }{A_{0}^{2}+B_{0}^{2}}\\ y_{P}\rightarrow \dfrac{-A_{0}d_{1}-B_{0}\left( C_{0}-d_{0}\sqrt{A_{0}^{2}+B_{0}^{2}}\right) }{A_{0}^{2}+B_{0}^{2}} \end{array} \end{aligned}$$

where \(d_{1}\) is an indeterminate introduced by the construction.

C11 Unconstrained point Point P is unconstrained

$$\begin{aligned} \begin{array}{l} x_{P}\rightarrow x_{0}\\ y_{P}\rightarrow y_{0} \end{array} \end{aligned}$$

where \(x_{0},y_{0}\) are indeterminates introduced by the construction.

C12 Line directed distance from one point Line L is directed distance \(d_{0}\) from point \(P_{0}\)

$$\begin{aligned} \begin{array}{l} A_{L}\rightarrow cos\left( \theta _{0}\right) \\ B_{L}\rightarrow sin\left( \theta _{0}\right) \\ C_{L}\rightarrow d_{0}-x_{0}cos\left( \theta _{0}\right) -y_{0}sin\left( \theta _{0}\right) \end{array} \end{aligned}$$

where \(\theta _{0}\) is an indeterminate introduced by the construction.

C13 Line angle to a line Line L is angle \(\theta _{0}\) from line \(L_{0}\)

$$\begin{aligned} \begin{array}{c} A_{L}\rightarrow A_{0}cos\left( \theta _{0}\right) -B_{0}sin\left( \theta _{0}\right) \\ B_{L}\rightarrow A_{0}sin\left( \theta _{0}\right) +B_{0}cos\left( \theta _{0}\right) \\ C_{L}\rightarrow d_{0} \end{array} \end{aligned}$$

where \(d_{0}\) is an indeterminate introduced by the construction.

C14 Unconstrained line Line L is unconstrained

$$\begin{aligned} \begin{array}{c} A_{L}\rightarrow A_{0}\\ B_{L}\rightarrow B_{0}\\ C_{L}\rightarrow C_{0} \end{array} \end{aligned}$$

where \(A_{0},B_{0},C_{0}\) are indeterminates introduced by the construction.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Todd, P. A Symbolic Dynamic Geometry System Using the Analytical Geometry Method. Math.Comput.Sci. 14, 693–726 (2020). https://doi.org/10.1007/s11786-020-00490-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-020-00490-0

Keywords

Mathematics Subject Classification

Navigation