KoG 5–2000/01 Blaˇzenka Divjak: Notes on Taxicab Geometry
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1 Introduction to taxicab geometry in space. Now we would like to take a taxi and drive as fast
è è
ê é ê as possible from the point A x1 é y1 to the point B x2 y2 . Let’s imagine an ideal city in which the streets run only To achieve this goal, it would be reasonable to measure the horizontally and vertically and city blocks have the same distance between points A and B. We can use Euclidean size. The intersections of streets correspond to the points (”school”) way to compute the distance. Then distance be-
5 KoG 5–2000/01 Blaˇzenka Divjak: Notes on Taxicab Geometry
tween A and B is Proposition 2 The Cartesian plane is a model C ë
ù 2
ô
ú é ë ê
é Le where Le is a collection of vertical (x a and
î ï ï
2 2 î 2
è è è
è ô
é ê ë ì í ê í ê ê d A B x2 x1 y2 y1 (1) non-vertical y ë ax b lines of C is an abstract ge- ometry. But this number show us the distance between A and B as a
bird flies and it is useless for us if we take a taxi. So we use
û ü Definition 3 An abstract geometry S é L is an incidence another measure so called taxicab distance which counts geometry if the number of blocks we would have to travel to get from A to B ï Obviously for taxicab distance between A and B the
following formula holds 1. Every two distinct points in S lie on a unique line. ï
î 2. There exits three points which do not lie on one line.
è
ê ë ð í ð ð í ð dT A é B x1 x2 y1 y2 (2)
According to the above description we could define the Proposition 4 The Cartesian plane is an incidence geom-
2 etry.
ñ ò ó taxicab geometry as the metric system é dT where points are represented by the ordered pairs of whole num-
bers. This definition is appropriate for student’s use in a Definition 5 A distance function onasetS is a function
ý þ ô ö way it will be described in section 3. d:S S such that for all A é B S For pure mathematical research the taxicab geometry is de-
2
è
ô
ê fined on and it will be introduced like this in the next 1. d A é B 0;
section.
è ¢
ê ë ë 2. d A é B 0 A B;
As it often happens, the approach which is not interesting
ï è
for pure mathematical research is more useful. è
ê ë é ê 3. d A é B d B A
2 2
ý ô þ ô 2 Approaches to taxicab geometry Proposition 6 The taxicab distance d : ô given by
There are three fundamental approaches to geometry: î
è
ê ë ð í ð ð í ð
dT A é B x1 x2 y1 y2 (3) è
1. synthetic approach, è
é ê ë é ê é where A ë x1 y2 and B x1 y2 is a distance function
2 ï
on ô 2. metric approach,
3. group approach. Definition 7 Let l be a line in an incidence geometry
û ü
S é L . Assume that there is a distance function d on S. ô A function f : l þ is a ruler (or coordinate system)forl if 2.1 Metric approach 1. f is a bijection; In the metric approach the concepts of distance and angle
measure are added to an incidence geometry. American
õ ö
2. A é B l
mathematician G. D. Birkhoff is founder of this approach
è è
(1932). è
ê í ê ð ë é ê ð f A f B d A B (4) Let we explain metric approach briefly (it is done in details
in [1]). (Ruler equation).
û ü Definition 1 An abstract geometry A consists of a set S Definition 8 An incidence geometry S é L together with
(set of points) and a collection L of non-empty subsets of S a distance function d satisfies the ruler postulate if every
ö û ü é (collection of lines) such that: line l S has a ruler. In this case we say M = S é L d is
a metric geometry.
ï
õ ö ö ö ö
ë ÷ ø 1. A é B S l L with A l and B l Proposition 9 The Cartesian plane with the taxicab dis- 2. Every line has at least two points. tance is a metric geometry.
6
KoG 5–2000/01 Blaˇzenka Divjak: Notes on Taxicab Geometry
ï
è è ö è ©
é ê ê ë ê
Proof. Ruler for vertical line x ë 0is f a y y 3. If D int ABC then
î
è è
ë é ê ê ë ï
Ruler for nonvertical line y ax b is f x y î
è © è © è ©
î ï
ê ë ê
m ABD ê m DBC m ABC (6)
è è
ð ê ê
1 a ð x
ù ü
2 û
é é Definition 18 A protractor geometry é d m is a
ô S L
é ú
Definition 10 The model T = é Le dT is called the
û ü é taxicab plane. Pasch geometry S é L d together with an angle measure m ï By using the distance function we define betweenness and betweenness allow us to define elementary figures such as Proposition 19 Assume that me is an angle measure for
segments, rays, angles and triangles. Euclidean metric geometry. Then me is an angle measure
ù ù
2 ï 2
ô ô
é ú é é é ú for the Taxicab plane é Le dT So, Le dT me
Definition 11 Bisbetween A and C if A,B and C are dis- is a protractor geometry.
û ü é tinct collinear points in the metric geometry S é L d and
if There is another concept of the angle measure in Taxicab ï
î plane as it is introduced in [2]. We will briefly describe it
è è è
ê é ê ë é ê d A é B d B C d A C (5) here. Remark 12 In the taxicab plane we can find three points
which are not collinear but which satisfy the equation Definition 20 A t-radian is an angle whose vertex is the
£
è è è
¤ center of a unit (taxicab) circle and which intercepts an
í í ê é é ê é é ê 5 . For example if we take A 1 é 1 B 0 0 C 2 1 £ θ the equation 5 ¤ is satisfied but these three point are not arc of length 1. The taxicab measure of a taxicab angle collinear. This shows why the definition of between re- is the number of t-radians subtended by the angle on the
quires collinear points. unit taxicab circle about the vertex.
û ü é
Definition 13 Let S é L d be a metric geometry and let It follows that a taxicab unit circle has 8 t-radians and so
ï ï
S1 ¥ S S1 is said to be convex if for every two points P, Q π
we can say that in taxicab geometry ë 4 This taxicab ï ö S1, the segment PQ is a subset of S1 angle measure has a nice application to parallax (the ap-
parent shift of an object due to the motion of the observer).
û ü é Definition 14 We say that a metric geometry S é L d The parallax is used for estimating the distance to a nearby
satisfies the plane separation axiom (PSA) if for every object. In the article is discovered that the taxicab method ö l L é there are two sets H1 and H2 of S such that yields the same formula commonly used in the Euclidean case with the exception that the taxicab formula is exact.
1. S-l=H1 ¦ H2;
2. H1 and H2 are disjoint and each is convex; Definition 21 A neutral geometry (or absolute geometry) is a protractor geometry which satisfies Side-Angle-Side
/ ï
ö § ¨ ö 0 3. If A H1 and B H2 then AB l ë Axiom (SAS).
Definition 15 A Pasch geometry is a metric geometry The Taxicab plane is not a neutral geometry. which satisfies PSA.
Proposition 16 The Taxicab geometry is a Pasch geome- 2.2 Synthetic approach try. In synthetic (or axiomatic) approach we have to decide what are the important properties of the geometry and then Definition 17 Let r be a fixed positive number. In a Pasch 0 define these properties axiomatically. This approach was geometry, an angle measure (or protractor) based on r is 0 first used in Euclid’s famous Elements but was introduced a function m from the set of all angles to the set of real A completely by D. Hilbert in 1899, who put several mathe- numbers such that
matical areas on axiomatic ground.
© ö è © We have to choose axioms first and in this some criterions 1. If ABC A then 0 m ABC ê r0;
should be obeyed ( correspondence to an intuitive picture,
þ í
2. If AB lies in the edge of the half plane H1 and if θ is richness of the theory and consistency). In the synthetic
θ
a real number with 0 r0 é then there is a unique approach, congruence axioms replace distance and angu-
þ
í
è ©
ö θ ë ray BC with C H1 and m ABC ê ; lar measure.
7 KoG 5–2000/01 Blaˇzenka Divjak: Notes on Taxicab Geometry
2.3 Group approach Proposition 23 The number of points on an ellipse E is 2a ï The group approach was introduced by F. Klein and A. Cayley and involved transformation group of a geometry. Then geometry studies those properties which are invariant Definition 24 For an ellipse E the halfaxis b is
under the group of motions.
ï
û è è ü ö
ð í ð é é ê é é ê 2 b ë max y2 y1 for all A x1 y2 B x2 y2 E (8) The taxicab metric in ô is an interesting example of non- Euclidean metric and it is a special case of the more gen- eral Minkowski metric where the defining unit circle is a Proposition 25 For an ellipse E the following holds
certain symmetric closed convex curve. It has been shown ï in [6] that the respective isometries in taxicab space form î
a ë b c (9) a subgroup of the whole space.
3 Taxicab in classroom 4 Applications of the taxicab geometry
Why to introduce taxicab in classroom? At the end, let us say a few facts about applications of the taxicab geometry in a real world problems. Let me cite some experts. In the real world the shortest distance between two places for trains, planes and even for plains, is not as a bird flies 1. B. Gr¨unbaum, J. Mycielski: ”Undergraduate courses (Euclidean distance). For example, trains want to take the on the foundations of geometry need examples dif- shortest route possible from one place to another. The lat- ferent from the traditional Euclidean plane in order to give students experience in dealing with axiomatic tice points with respect to taxicab geometry would be the systems.” ([7]) depots or turns of the track. We want to consider another variable to travel along taxicab distance, so called taxi- 2. M. Artigue: ”...much of our knowledge remains cab time. But sometimes it is not good enough, because strongly contextual, that is, dependent on the situ- the shortest route is selected thus minimizing taxicab dis- ations from which it arises.” tance and minimizing costs. The minimal taxicab distance, ”...learning in mathematics is not a continuous pro- time or costs can be determined by the same method de- cess, (it is necessity sometimes to make) breaks with scribed in [9]. earlier knowledge and models of thought.” ( [8]) Let us explain another natural problem for taxicab geom- etry. A taxi company has a number of stations across the The main problem in introducing geometrical systems dif- city. In order to dispatch the calls, the company want to ferent from Euclidean on technical faculties is a pro- know for any point in the city, which station is the closest, gram frame. In that case solution is mathematical essay so that the taxi can get there as fast as possible. Obviously, or diploma work, but then with special emphasis on appli- our task is to find the locus of points which are equidis- cations aspects of taxicab geometry (see [2] and[4]). tant to their two closest taxi stations. These boundaries On mathematical faculties it is recommended for students will tessellate the plane in regions. In Euclidean geometry, to investigate some theoretical aspects, such as circles (as the corresponding graph is called Vornoi diagram and is it is done in [5]) or other curves of second order in taxicab widely used in computational geometry and pattern recog- geometry (see [3]). It will be easier, and for this maybe nition (see [10]). This task can be a interesting diploma more motivating, to take ”student’s version” of taxicab ge- work on mathematical or technical oriented faculty (a Java 2 ometry (points are from ò ). program can be demanded). In addition some results of such investigation on ellipse A city planning makes also attractive use of taxicab ge- will be mentioned. ometry. From the historical point of view there are two
types of cities: organically grown cities and planned cities
è è
í ê é é ê Definition 22 Let F1 c é 0 F2 c 0 be two point in (see Fig. 1 and Fig. 2). Organically grown cities develop
2
ò
ñ é dT ó (focuses). The ellipse E is the set of points slowly, tend to plan around preserving their natural beauty
2
è ò ê T x é y from such that and have unstructured design with curved shapes and spo-
radic placements of streets, buildings and landscape (early
ï î
è è ö ò
é ê é ê ð ë é ð dT F1 T dT F1 T a a (7) European cities).
8 KoG 5–2000/01 Blaˇzenka Divjak: Notes on Taxicab Geometry
planning city and the traversed routes between and through cities. ”Taxicab geometry is a great model for the artifi- cially grid planned world that man has created” (cite [4]).
References
[1] Millman R. S.,Parker G.D.; Geometry: A Metric Approach with Models, Springer-Verlag, New York (1991). [2] Thompson K., Dray T.; Taxicab Angles and Trigonometry, Pi Mu Epsilon Journal (2000) (to apear) Figure 1: Aachen - the example of a “spider” city. [3] Prevost F.J.; The conic sections in taxicab geome- try: Some investigations for high school students, The Mathematics Theachers, Reston, Apr (1998).
[4] http://www2.gvsu.edu/˜vanbelkj/Project.html [5] Tian S., So S., Chen G.; Concerning circles in taxicab geometry, International Journal of Mathematical Ed- ucation in Science and Technology, Sep/Oct (1997). [6] Greenberg M.;Euclidean and Non-Euclidean Geom- etry, Freeman, San Francisco (1980). [7] Gr¨unbaum B., Mycielski J.; Some models of Plane Geometries,AMM, Vol 97, No9, Nov (1990) pp.839- 847. [8] Artigue M.; The Teaching and Learning of Mathe- Figure 2: Baltimore - the example of a planned city. matics at the University Level: Crucial Questions for Contemporary Reasearh in Education, Notices of the The growth and plans of planned cities are on ”high levels” AMS, Dec (1999). coordinated with a population growth (predicted mathe- matically on the basis of cohort survival method). The pat- [9] Hesse R., Gene W.; Applied Management Science A terns in the planned cities are more vertical and horizontal Quick & Dirty (and more suitable for application of taxicab geometry). In Approach, Science Research Associates, Inc. (1980), the 1960s the central focus of city planning moved from pp. 96-99. artistic beauty of the city towards location and efficiency. [10] Graham R., Yao F.; A Whirlwind of Computational But even for the curved streets we can set up models such Geometry, The American Mathematical Monthly, as the shortest rout problem or Vornoi diagrams. The Vol. 97, No. 8, Oct (1990), pp. 687-702. curved streets can be represent as a straight edge form with the distance used as edge weighs and the street intersec-
tions as the nodes. Besides a distance we have to consider
P F 2 5 B : 8 × B 3 7 B traffic volume, speed limits, traffic signals and so on.
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Additionally, the street patterns tells a lot about the city’s
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We can conclude by noticing that parts of city planning in-
D @ & / E * : G & J K $ " M N volve the use of geometric models and mathematical mod- els in general. Taxicab geometry is used in many ways for
9