The Central Idea
A geometry is shaped not only by the points and figures being studied but also by the rule used to measure distance.
In ordinary coordinate geometry, the distance from \(P=(x_1,y_1)\) to \(Q=(x_2,y_2)\) is
\[ d_2(P,Q)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. \]This is Euclidean distance. It measures the length of the straight segment joining the points.
In taxicab geometry, the distance is instead
\[ \boxed{d_1(P,Q)=|x_2-x_1|+|y_2-y_1|}. \]The same two points can therefore have different distances depending on the metric being used.
Why Change the Distance Rule?
Euclidean distance is natural when direct straight-line motion is possible. Many real systems impose constraints that make another rule more useful.
- City travel may be constrained by a street grid.
- Robots may move only along permitted directions.
- Networks may require travel along edges rather than through open space.
- Some directions may be more expensive than others.
- Computer systems may compare data by coordinate differences rather than physical length.
- Games may allow a piece to move according to special movement rules.
A useful distance rule should still behave like distance in several basic ways. Those requirements lead to the concept of a metric.
What Is a Metric?
Let \(X\) be a set. A metric on \(X\) is a function
\[ d:X\times X\to\mathbb{R} \]that assigns a distance \(d(P,Q)\) to every pair of points \(P,Q\in X\) and satisfies four properties.
- Nonnegativity: \(d(P,Q)\ge 0\).
- Identity of indiscernibles: \(d(P,Q)=0\) exactly when \(P=Q\).
- Symmetry: \(d(P,Q)=d(Q,P)\).
- Triangle inequality: \(d(P,R)\le d(P,Q)+d(Q,R)\).
A set together with a metric is called a metric space.
Nonnegativity
A distance cannot be negative:
\[ d(P,Q)\ge0. \]A proposed rule that can produce a negative value cannot be a metric.
Zero Distance Means the Same Point
A metric must satisfy
\[ d(P,Q)=0\iff P=Q. \]Different points must have positive distance from each other. This condition prevents the metric from collapsing distinct points together.
Symmetry
The distance from \(P\) to \(Q\) must equal the distance from \(Q\) to \(P\):
\[ d(P,Q)=d(Q,P). \]Travel time on a one-way road network may fail this property, so a directed travel-cost function is not automatically a metric even if it is useful for another purpose.
The Triangle Inequality
The triangle inequality says that taking an intermediate stop cannot produce a route shorter than the distance assigned directly between the endpoints:
\[ \boxed{d(P,R)\le d(P,Q)+d(Q,R)}. \]This is one of the most important structural requirements for a metric.
Taxicab Distance
For points
\[ P=(x_1,y_1),\qquad Q=(x_2,y_2), \]the taxicab distance is
\[ \boxed{d_T(P,Q)=|x_2-x_1|+|y_2-y_1|}. \]This metric is also called the Manhattan distance, rectilinear distance, or the \(L^1\) distance.
Example: A Simple Taxicab Distance
Find the taxicab distance between
\[ P=(1,2) \]and
\[ Q=(6,5). \]The horizontal change is
\[ |6-1|=5, \]and the vertical change is
\[ |5-2|=3. \]Therefore,
\[ \boxed{d_T(P,Q)=5+3=8}. \]Compare with Euclidean Distance
For the same points \(P=(1,2)\) and \(Q=(6,5)\), the Euclidean distance is
\[ d_E(P,Q)=\sqrt{(6-1)^2+(5-2)^2}. \]Thus,
\[ d_E(P,Q)=\sqrt{25+9}=\sqrt{34}\approx5.83. \]The taxicab distance is \(8\), while the Euclidean distance is about \(5.83\). Neither answer is inherently wrong. Each belongs to a different metric.
Why Absolute Values Appear
Horizontal and vertical changes may be positive or negative, but traveled distance must be nonnegative. Absolute values convert coordinate changes into positive travel amounts.
For example, from \((5,7)\) to \((2,1)\),
\[ |2-5|=3 \]and
\[ |1-7|=6. \]Therefore,
\[ d_T=3+6=9. \]Taxicab Geometry as Grid Travel
Imagine a city whose streets run only north-south and east-west. To travel from one intersection to another, a traveler must accumulate enough horizontal movement and enough vertical movement to match the coordinate differences.
If the required horizontal change is \(a\) and the required vertical change is \(b\), every shortest grid route has length
\[ |a|+|b|. \]Shortest Paths Need Not Be Unique
In the Euclidean plane, two distinct points determine one straight segment, and that segment is the unique shortest path between them.
In taxicab geometry, shortest paths are often not unique.
From \((0,0)\) to \((3,2)\), a traveler may move three units east and then two units north, or two units north and then three units east, or alternate those movements in many ways. Every route that never wastes horizontal or vertical travel has total length
\[ 3+2=5. \]Monotone Shortest Paths
Suppose \(Q\) lies northeast of \(P\). A taxicab path from \(P\) to \(Q\) is shortest if it accumulates exactly the required eastward and northward movement without unnecessary reversal.
Any detour that moves west and later compensates by moving farther east adds extra length. The same is true for unnecessary vertical reversal.
Counting Grid Shortest Paths
On an integer grid, combinatorics can count shortest paths.
To move from \((0,0)\) to \((3,2)\), a shortest path uses exactly three east steps and two north steps. These five moves can be arranged in
\[ \binom{5}{2}=10 \]different orders.
This connects geometry with discrete counting.
What Is a Circle in a Metric?
A circle is not fundamentally defined by roundness. In a metric space, a circle of radius \(r\) centered at \(C\) is the set of points \(P\) satisfying
\[ \boxed{d(C,P)=r}. \]The shape of the circle therefore depends on the distance rule.
Taxicab Circles
Consider the taxicab circle of radius \(r\) centered at the origin. Its points satisfy
\[ |x|+|y|=r. \]For \(r=4\), the intercepts are
\[ (4,0),\quad(0,4),\quad(-4,0),\quad(0,-4). \]Connecting the appropriate linear pieces produces a diamond-shaped boundary in ordinary Cartesian drawing.
The Circle Is Not “Really” a Diamond
Calling the taxicab circle a diamond describes how it looks when drawn using ordinary Euclidean coordinates on paper. Intrinsically, it is the set of all points at the same taxicab distance from its center.
This distinction is important throughout non-Euclidean and alternative geometry: a familiar drawing may use one geometry to represent another.
Taxicab Disks
A closed metric disk of radius \(r\) centered at the origin consists of the points satisfying
\[ |x|+|y|\le r. \]The corresponding open disk satisfies
\[ |x|+|y|Euclidean and Taxicab Circles Compared
A Euclidean circle centered at the origin satisfies
\[ x^2+y^2=r^2. \]A taxicab circle satisfies
\[ |x|+|y|=r. \]Both are circles according to the same abstract definition—points at fixed distance from a center—but the different metrics produce different shapes.
Taxicab Distance in Higher Dimensions
In \(n\)-dimensional space, taxicab distance generalizes naturally. For
\[ P=(p_1,p_2,\ldots,p_n) \]and
\[ Q=(q_1,q_2,\ldots,q_n), \]define
\[ \boxed{d_1(P,Q)=\sum_{i=1}^{n}|p_i-q_i|}. \]The subscript \(1\) connects this metric to the \(L^1\) norm.
Distance from a Norm
In a vector space, many metrics arise from a norm. A norm measures the size of a vector. If \(\|v\|\) is a norm, then a corresponding distance is
\[ d(P,Q)=\|P-Q\|. \]Euclidean and taxicab distance are both examples of this construction.
The Euclidean \(L^2\) Norm
For a vector \(v=(x,y)\), the Euclidean norm is
\[ \|v\|_2=\sqrt{x^2+y^2}. \]Thus,
\[ d_2(P,Q)=\|P-Q\|_2. \]This is why ordinary Euclidean distance is also called the \(L^2\) distance.
The Taxicab \(L^1\) Norm
For \(v=(x,y)\), the \(L^1\) norm is
\[ \|v\|_1=|x|+|y|. \]Therefore,
\[ d_1(P,Q)=\|P-Q\|_1. \]The \(L^p\) Family
The \(L^1\) and \(L^2\) distances belong to a larger family. For finite coordinate vectors and \(p\ge1\), define
\[ \boxed{d_p(P,Q)=\left(\sum_{i=1}^{n}|p_i-q_i|^p\right)^{1/p}}. \]Important cases include
- \(p=1\): taxicab or Manhattan distance;
- \(p=2\): Euclidean distance;
- large \(p\): increasing emphasis on the largest coordinate difference.
Why We Usually Require \(p\ge1\)
For \(p\ge1\), the \(L^p\) formula satisfies the triangle inequality and defines a genuine metric on finite-dimensional coordinate space.
If \(0
The \(L^\infty\) or Chebyshev Metric
Another important distance is determined only by the largest coordinate difference:
\[ \boxed{d_\infty(P,Q)=\max_i|p_i-q_i|}. \]In the plane,
\[ d_\infty(P,Q)=\max\{|x_2-x_1|,|y_2-y_1|\}. \]This is called the Chebyshev distance or maximum metric.
Example: Chebyshev Distance
For \(P=(1,2)\) and \(Q=(6,5)\), the coordinate differences are \(5\) and \(3\). Therefore,
\[ \boxed{d_\infty(P,Q)=\max\{5,3\}=5}. \]Compare the three metrics:
\[ d_1=8, \] \[ d_2=\sqrt{34}\approx5.83, \] \[ d_\infty=5. \]Chebyshev Distance and King Moves
On a chessboard, a king can move one square horizontally, vertically, or diagonally in a single move. Ignoring obstacles, the minimum number of king moves between two squares is the larger of the horizontal and vertical separations.
That movement rule is modeled naturally by the Chebyshev metric.
Circles in the \(L^\infty\) Metric
A circle of radius \(r\) centered at the origin satisfies
\[ \max\{|x|,|y|\}=r. \]In ordinary Cartesian drawing, this boundary is a square with sides parallel to the coordinate axes and vertices
\[ (r,r),\quad(r,-r),\quad(-r,r),\quad(-r,-r). \]Three Unit Circles
The unit circle means the set of points distance \(1\) from the origin. Under three familiar metrics:
- \(L^1\): \(|x|+|y|=1\), a diamond in Euclidean drawing;
- \(L^2\): \(x^2+y^2=1\), the familiar round circle;
- \(L^\infty\): \(\max\{|x|,|y|\}=1\), an axis-aligned square.
This comparison makes the influence of a metric visible immediately.
A Useful Inequality among the Three Metrics
For points in finite-dimensional coordinate space,
\[ d_\infty(P,Q)\le d_2(P,Q)\le d_1(P,Q). \]In the plane, a slightly sharper upper comparison is
\[ d_1(P,Q)\le\sqrt{2}\,d_2(P,Q). \]These inequalities show that the metrics assign different numerical values while remaining systematically related.
Weighted Taxicab Distance
Sometimes movement in one direction costs more than movement in another. Positive weights can model that difference:
\[ \boxed{d(P,Q)=a|x_2-x_1|+b|y_2-y_1|}, \]where \(a>0\) and \(b>0\).
If vertical travel costs twice as much as horizontal travel, one might use
\[ d(P,Q)=|x_2-x_1|+2|y_2-y_1|. \]Example: Weighted Distance
Let
\[ d(P,Q)=2|x_2-x_1|+5|y_2-y_1|. \]For \(P=(1,1)\) and \(Q=(4,3)\),
\[ |4-1|=3, \]and
\[ |3-1|=2. \]Therefore,
\[ d(P,Q)=2(3)+5(2)=16. \]Anisotropic Geometry
A geometry is anisotropic when different directions behave differently. Weighted metrics provide a simple example because motion along one coordinate direction may be more expensive than motion along another.
Euclidean distance is rotationally symmetric, but a weighted coordinate metric generally is not.
The Discrete Metric
On any set \(X\), define
\[ d(P,Q)= \begin{cases} 0, & P=Q,\\ 1, & P\ne Q. \end{cases} \]This is the discrete metric.
Every pair of distinct points is exactly one unit apart. The rule may seem unusual geometrically, but it satisfies all four metric axioms.
Why the Discrete Metric Satisfies the Triangle Inequality
If \(P=R\), then \(d(P,R)=0\), so the inequality is immediate.
If \(P\ne R\), then \(d(P,R)=1\). At least one of \(d(P,Q)\) or \(d(Q,R)\) must be \(1\), so
\[ d(P,Q)+d(Q,R)\ge1=d(P,R). \]Thus the triangle inequality holds.
Graph Distance
A graph consists of vertices connected by edges. If each edge has length \(1\), the graph distance between two vertices is the minimum number of edges in a path connecting them.
More generally, if edges carry positive weights, the distance can be defined as the minimum total edge weight among all connecting paths.
Why Graph Distance Is Geometric
Geometry does not require a continuous plane. A network can have its own meaningful notions of distance, shortest path, neighborhood, and center.
Road networks, communication systems, airline routes, electrical networks, and social connections can all be studied using graph-based metrics.
Shortest-Path Metrics
The graph example illustrates a broader construction. If movement is restricted to an allowed collection of paths and each path has a length or cost, define the distance between two points as the infimum—or, when attained, the minimum—of the lengths of admissible paths joining them.
Many geometric distances arise from this shortest-path principle.
Great-Circle Distance Is Another Metric
Earlier in this section, we studied distance on a sphere. The shortest surface distance between two points on a sphere is measured along a great-circle arc.
That distance is not ordinary straight-line Euclidean distance through the surrounding three-dimensional space. It is an intrinsic metric on the sphere.
Intrinsic Distance versus Ambient Distance
Suppose two points lie on a curved surface embedded in space. Two natural distances may be considered:
- the straight-line distance through the surrounding space;
- the shortest distance constrained to remain on the surface.
These generally differ. The second is intrinsic to the surface and connects alternative metrics with the earlier lessons on geodesics and curved geometry.
Metric Balls
Given a metric \(d\), a center \(C\), and a radius \(r>0\), the open ball is
\[ B_r(C)=\{P:d(C,P)The shapes of these balls depend on the metric.
Neighborhoods Depend on Distance
A metric determines what it means for one point to be near another. Metric balls therefore provide a precise language for neighborhoods.
This becomes important in later studies of continuity, convergence, and topology, where local behavior is described using arbitrarily small neighborhoods.
Different Metrics Can Produce the Same Local Notion of Nearness
Two metrics may assign different numerical distances yet still generate the same open sets and therefore the same topology.
In finite-dimensional Euclidean coordinate space, the usual \(L^1\), \(L^2\), and \(L^\infty\) metrics are different metrics but produce the same standard topology.
This is an advanced idea, but it reinforces an important distinction: numerical distance and qualitative neighborhood structure are related but not identical concepts.
Distance Is Not the Same as Coordinate Difference
A coordinate difference such as
\[ x_2-x_1 \]can be negative and records directed change along one axis. A metric distance must be nonnegative and usually combines information from all relevant coordinates.
Distance Is Not the Same as Displacement
In vector language, displacement from \(P\) to \(Q\) is the vector
\[ Q-P. \]Distance is a scalar quantity obtained by measuring that displacement under some metric or norm.
Different norms can assign different lengths to the same displacement vector.
Translation-Invariant Metrics
Euclidean, taxicab, and Chebyshev distances satisfy
\[ d(P+A,Q+A)=d(P,Q). \]Moving both points by the same translation does not change their distance. This is why these metrics depend only on the difference \(P-Q\).
Rotational Symmetry Can Change
Euclidean distance is unchanged by every ordinary rotation. Taxicab distance is not invariant under arbitrary rotations of the coordinate plane.
For example, the vectors \((1,0)\) and \((1/\sqrt{2},1/\sqrt{2})\) have the same Euclidean length \(1\), but their taxicab lengths are
\[ 1 \]and
\[ \sqrt{2}, \]respectively.
Perpendicular Bisectors Can Change Shape
In Euclidean geometry, the set of points equidistant from two distinct points is a straight perpendicular bisector.
Under another metric, the equidistant set can have a different shape and may contain several linear pieces or even regions in special configurations.
The definition remains
\[ d(P,A)=d(P,B), \]but the distance formula changes the resulting locus.
Voronoi Regions Depend on the Metric
A Voronoi region contains the points closer to one site than to competing sites. Because “closer” depends on the metric, changing the metric changes Voronoi boundaries and cells.
Taxicab and weighted metrics are therefore useful in location problems where travel constraints or costs differ from straight-line distance.
Alternative Metrics in Data Analysis
Data points can be treated as vectors with many coordinates. A metric can then measure how different two observations are.
Different metrics emphasize different kinds of discrepancy:
- \(L^1\) adds absolute coordinate differences;
- \(L^2\) emphasizes larger differences through squaring;
- \(L^\infty\) focuses entirely on the largest coordinate difference;
- weighted metrics allow some variables to matter more than others.
Scaling Matters in Applied Metrics
Suppose one coordinate is measured in dollars and another in millimeters. Directly combining the raw coordinate differences may give a misleading distance because the numerical scales differ greatly.
In applications, coordinates may need normalization or meaningful weights before a metric represents the intended notion of similarity or cost.
A Metric Is a Modeling Choice
Mathematics can verify whether a proposed rule satisfies the metric axioms, but the choice of metric also depends on what the model is intended to represent.
A mathematically valid metric can still be a poor model of a particular physical or operational system.
Example: Comparing Three Metrics
Let
\[ P=(-2,1),\qquad Q=(4,5). \]The coordinate differences have magnitudes
\[ |4-(-2)|=6, \]and
\[ |5-1|=4. \]Therefore,
\[ d_1(P,Q)=6+4=10, \] \[ d_2(P,Q)=\sqrt{6^2+4^2}=\sqrt{52}=2\sqrt{13}, \]and
\[ d_\infty(P,Q)=6. \]Example: A Taxicab Circle
Describe the taxicab circle of radius \(3\) centered at \((2,-1)\).
The defining equation is
\[ \boxed{|x-2|+|y+1|=3}. \]Its four Euclidean-drawing vertices are
\[ (5,-1),\quad(-1,-1),\quad(2,2),\quad(2,-4). \]Example: A Chebyshev Circle
Describe the \(L^\infty\) circle of radius \(2\) centered at \((1,3)\).
Its defining equation is
\[ \boxed{\max\{|x-1|,|y-3|\}=2}. \]In an ordinary Cartesian drawing, this is the boundary of the square
\[ -1\le x\le3, \]and
\[ 1\le y\le5. \]Example: Verifying a Proposed Metric Rule
Consider the rule on the real line
\[ d(x,y)=|x-y|. \]It is nonnegative, equals zero exactly when \(x=y\), and is symmetric. The ordinary absolute-value triangle inequality gives
\[ |x-z|\le|x-y|+|y-z|. \]Therefore, the rule is a metric.
Example: A Rule That Is Not a Metric
Consider
\[ d(x,y)=(x-y)^2 \]on the real line. This rule is nonnegative, symmetric, and zero exactly when \(x=y\). But the triangle inequality can fail.
Let \(x=0\), \(y=1\), and \(z=2\). Then
\[ d(0,2)=4, \]while
\[ d(0,1)+d(1,2)=1+1=2. \]Since \(4\nleq2\), the rule is not a metric.
Example: A Rule That Fails Identity
Suppose
\[ d((x_1,y_1),(x_2,y_2))=|x_2-x_1|. \]Points with the same \(x\)-coordinate but different \(y\)-coordinates have distance zero. For example,
\[ d((3,1),(3,9))=0. \]Because distinct points can have zero distance, this rule is not a metric on the full plane.
Common Error: Using the Euclidean Formula Automatically
If a problem specifies taxicab distance, the Pythagorean distance formula is not the relevant rule.
For \((0,0)\) and \((3,4)\),
\[ d_E=5, \]but
\[ d_T=3+4=7. \]Always identify the metric before calculating.
Common Error: Forgetting Absolute Values
The expression
\[ (x_2-x_1)+(y_2-y_1) \]is not the taxicab distance because positive and negative coordinate changes could cancel.
Taxicab distance requires
\[ |x_2-x_1|+|y_2-y_1|. \]Common Error: Thinking a Circle Must Look Round
A circle is defined using equal distance from a center. The visual shape of that set depends on the metric.
Roundness is a feature of Euclidean circles, not part of the abstract metric definition of a circle.
Common Error: Assuming Shortest Paths Are Unique
Euclidean experience can create the expectation that two points always have one shortest path. Taxicab geometry shows that this is not a universal property.
Many distinct monotone paths can have the same minimum taxicab length.
Common Error: Treating Every Cost Function as a Metric
A useful cost rule need not satisfy the metric axioms. One-way travel costs, for example, can fail symmetry. Other formulas can fail the triangle inequality or allow distinct points to have zero distance.
“Useful measure” and “metric” are not interchangeable terms.
Common Error: Confusing the Metric with the Drawing
A taxicab circle is often drawn on ordinary Euclidean graph paper. The paper provides the visual coordinate representation; the taxicab formula provides the intrinsic distance.
The Euclidean appearance of the drawing does not replace the metric being studied.
A Reliable Procedure for Metric-Distance Problems
- Identify the metric explicitly.
- Write its formula before substituting coordinates.
- Compute each coordinate difference.
- Apply absolute values, powers, weights, or maxima as required.
- Combine the pieces according to the metric.
- Check that the result is nonnegative.
- Interpret the value in the geometry or model being used.
A Reliable Procedure for Testing a Proposed Metric
- Check nonnegativity.
- Check whether distance zero occurs exactly for identical points.
- Check symmetry.
- Check the triangle inequality.
- If one axiom fails, provide a specific counterexample.
- If all four hold, justify each one rather than relying only on examples.
Full Example: Comparing Movement Models
Suppose a robot moves from
\[ P=(2,-3) \]to
\[ Q=(-4,5). \]The coordinate differences have magnitudes
\[ |-4-2|=6 \]and
\[ |5-(-3)|=8. \]If the robot can move directly in any direction, Euclidean distance gives
\[ d_2=\sqrt{6^2+8^2}=10. \]If it can move only horizontally and vertically, taxicab distance gives
\[ d_1=6+8=14. \]If a movement system allows simultaneous horizontal and vertical progress at equal per-step speed, the Chebyshev model gives
\[ d_\infty=\max\{6,8\}=8. \]The coordinates are unchanged. Only the movement model—and therefore the metric—has changed.
Full Example: Weighted Urban Travel
Suppose east-west travel costs \(3\) units per block and north-south travel costs \(5\) units per block. Let
\[ P=(1,4),\qquad Q=(7,-2). \]The horizontal separation is
\[ |7-1|=6, \]while the vertical separation is
\[ |-2-4|=6. \]The weighted distance is
\[ d(P,Q)=3(6)+5(6)=48. \]This quantity models cost rather than ordinary physical length, but because the weights are positive it still defines a metric of weighted \(L^1\) type.
Practice 1: Taxicab Distance
Find the taxicab distance between
\[ P=(-3,4),\qquad Q=(5,-2). \]Practice 2: Compare \(L^1\), \(L^2\), and \(L^\infty\)
For
\[ P=(0,0),\qquad Q=(8,6), \]calculate \(d_1\), \(d_2\), and \(d_\infty\).
Practice 3: Taxicab Circle
Write the equation of the taxicab circle of radius \(5\) centered at \((-2,3)\), and identify its four Euclidean-drawing vertices.
Practice 4: Chebyshev Circle
Write the equation of the \(L^\infty\) circle of radius \(3\) centered at \((4,-1)\).
Practice 5: Weighted Metric
Let
\[ d(P,Q)=4|x_2-x_1|+2|y_2-y_1|. \]Find the distance between \((1,5)\) and \((-2,-1)\).
Practice 6: Discrete Metric
Under the discrete metric, find the distance between two distinct points \(A\) and \(B\). Then find \(d(A,A)\).
Practice 7: Is It a Metric?
On the real line, consider
\[ d(x,y)=2|x-y|. \]Does this define a metric? Explain.
Practice 8: Find the Failed Axiom
On the real line, consider
\[ d(x,y)=x-y. \]Identify at least one metric axiom that fails.
Practice 9: Shortest Grid Paths
On an integer grid, how many shortest paths from \((0,0)\) to \((4,3)\) use only unit east and north moves?
Practice 10: Modeling Choice
A delivery vehicle must remain on a rectangular street grid. Explain why Euclidean distance can underestimate the travel distance and identify a more appropriate idealized metric.
Practice Answers
Practice 1
The coordinate differences are \(8\) and \(6\), so
\[ \boxed{d_1=8+6=14}. \]Practice 2
\[ d_1=8+6=14, \] \[ d_2=\sqrt{8^2+6^2}=10, \] \[ \boxed{d_\infty=8}. \]Practice 3
The circle is
\[ \boxed{|x+2|+|y-3|=5}. \]Its four vertices in the ordinary coordinate drawing are
\[ (3,3),\quad(-7,3),\quad(-2,8),\quad(-2,-2). \]Practice 4
\[ \boxed{\max\{|x-4|,|y+1|\}=3}. \]Practice 5
The coordinate differences have magnitudes \(3\) and \(6\). Thus,
\[ d=4(3)+2(6)=24. \]Therefore,
\[ \boxed{d=24}. \]Practice 6
Distinct points have distance \(1\), while a point has distance \(0\) from itself:
\[ \boxed{d(A,B)=1,\qquad d(A,A)=0}. \]Practice 7
Yes. Multiplying the ordinary metric \(|x-y|\) by the positive constant \(2\) preserves nonnegativity, identity, symmetry, and the triangle inequality. Therefore,
\[ \boxed{d(x,y)=2|x-y|} \]is a metric.
Practice 8
Nonnegativity fails because \(x-y\) can be negative. Symmetry also fails in general because
\[ x-y\ne y-x. \]Therefore the rule is not a metric.
Practice 9
A shortest path uses four east moves and three north moves, for seven moves total. Choose the positions of the three north moves:
\[ \boxed{\binom{7}{3}=35}. \]Practice 10
Euclidean distance measures the straight segment through open space, which may not be an allowed route on the street grid. An idealized taxicab metric measures the required horizontal plus vertical travel:
\[ \boxed{d_T=|\Delta x|+|\Delta y|}. \]Real road networks may require an even more detailed weighted graph metric, but taxicab distance captures the basic grid constraint.
What to Remember
- A metric is a rule for distance satisfying four specific axioms.
- The axioms are nonnegativity, identity, symmetry, and the triangle inequality.
- Euclidean distance is only one possible metric.
- Taxicab distance in the plane is \(|\Delta x|+|\Delta y|\).
- Taxicab distance is also called Manhattan, rectilinear, or \(L^1\) distance.
- Euclidean distance is the \(L^2\) distance.
- Chebyshev distance is the \(L^\infty\) or maximum metric.
- For finite coordinate vectors, \(L^p\) formulas define metrics for \(p\ge1\).
- A circle is the set of points a fixed metric distance from a center.
- Taxicab circles appear diamond-shaped in ordinary Cartesian drawings.
- Chebyshev circles appear square in ordinary Cartesian drawings.
- Changing the metric can change shortest paths, circles, bisectors, and Voronoi regions.
- Taxicab shortest paths are often nonunique.
- Weighted metrics can model unequal movement costs.
- The discrete metric places every two distinct points one unit apart.
- Graph distance measures shortest travel through a network.
- Surface geodesic distance is another example of a non-Euclidean metric.
- Different metrics can assign different numbers while still producing the same topology.
- A metric is both a mathematical structure and, in applications, a modeling choice.
Where We Go Next
Alternative metrics showed that geometry can change when the rule for distance changes, even if the same points remain in place. The next lesson changes another familiar assumption: it considers geometries whose sets of points themselves are finite.
In Finite Geometries, we will study incidence systems with finitely many points and lines, including finite affine and projective planes. These examples reveal that rich geometric structure does not require a continuous plane or infinitely many points.