Orienting Yourself: The Use of Coordinates

Historical Context and Global Roots of Coordinate Systems

Definition of a Coordinate System: A structured framework, analogous to grid lines on a map or graph paper, used to describe the precise physical locations of points or objects through numerical values.
Roots in Bhārat: Grid-based thinking and the geometry required for spatial positioning have deep historical roots in India. * Sindhu-Sarasvatī Civilisation: The first systematic application of grids occurred thousands of years ago on a massive urban scale. City streets were constructed with precision in North–South and East–West directions, positioned at uniform distances of approximately $10\,m$ apart. This allowed merchants to locate shops or warehouses by counting units from the city center. * Baudhāyana (c. 800 CE): Utilized East–West and North–South lines for complex geometric constructions. He developed the Baudhāyana–Pythagoras Theorem, providing the foundations for coordinate geometry.
Navigational Importance and Global Evolution: * Ujjayinī: Recognized as early as the $4^{th}$ century BCE in the early Siddhāntas as the central longitude meridian from which all other locations were measured. * Ptolemy (c. 150 BCE): Building on the work of Hipparchus, he described the latitudes and longitudes of thousands of locations, including Ujjayinī, which he referred to as ‘Ozine’. * Āryabhaᅩa (c. 499 CE): Revolutionized calculations by replacing Greek ‘chords’ with ‘sines’. He mapped the sky using Celestial Coordinates, measuring distances from the ecliptic (the sun's path). * Brahmagupta (c. 628 CE): Formalized the concept and usage of zero and negative numbers as algebraic entities. This was essential for the creation of modern coordinate systems where the ‘origin’ is zero and negative axes represent values less than zero. * Translation to Arabic Geography: Brahmagupta’s work entered the Arabic world via the Sindhind. The Ujjayinī meridian became ‘Arin’, the zero-longitude reference for early Arabic maps. * Al-Bīũnī (c. 1000 CE): Studied the Siddhāntas to calculate city coordinates across Asia using Indian trigonometry. He also perfected the astrolabe, a handheld device for sailors to determine coordinates via the stars. * Ömar Khayyām (c. 1100 CE): An expert in the Indian decimal system and algebraic formalism, he was the first to solve algebraic problems by interpreting them as geometry within a coordinate plane.
European Formalization: * Fermat (1636 CE) and René Descartes (1637 CE): Formalized the principle that any point in a 2-D plane can be defined by two numbers representing distances from two perpendicular axes. This bridged the gap between algebra and geometry, allowing shapes to be described by equations.

Practical Application: Shalini and Reiaan’s Room Map

Scenario: Reiaan is a student who is navigating a new home in a new city. Because he cannot see, his sister Shalini (who just completed Grade 9) uses coordinate geometry to help him visualize their room.
Methodology: Shalini created a tactile map of the room floor using a rectangular grid. * Materials: Pins to mark locations, and threads or thick wool to connect corners of objects. * Scale: $1\,cm : 1\,foot$ .
Limitations of the Map: The map represents the floor layout; therefore, it cannot represent the vertical position of windows (which are situated above the floor level).

The 2-D Cartesian Coordinate System

Fundamental Components: * Axes: Two lines intersecting at right angles ( $90^{\circ}$ ). The horizontal line is the x-axis, and the vertical line is the y-axis. * Origin (O): The point where the axes intersect, with coordinates $(0, 0)$ . * Coordinate Axes: The plural form of ‘axis’; they act as the reference for locating points using equal units of distance from O.
Directional Conventions: * Positive: Distances to the right of O (along the x-axis) or upwards from O (along the y-axis). * Negative: Distances to the left of O or downwards from O.
Notation: A point is written as $P = (x, y)$ or simply $P(x, y)$ . * x-coordinate: The perpendicular distance from the y-axis. * y-coordinate: The perpendicular distance from the x-axis.
Axis-Specific Points: * Points on the x-axis have the form $(x, 0)$ . If x > 0, the point is right of the origin; if x < 0, it is left. * Points on the y-axis have the form $(0, y)$ . If y > 0, the point is above the origin; if y < 0, it is below.

The Cartesian Plane and Quadrants

Definition: The plane containing the axes is called the Cartesian plane, the coordinate plane, or the xy-plane.
Division of Space: The axes divide the plane into four parts known as quadrants, numbered I through IV: * Quadrant I: Positive x, Positive y ( $+ , +$ ). * Quadrant II: Negative x, Positive y ( $- , +$ ). * Quadrant III: Negative x, Negative y ( $- , -$ ). * Quadrant IV: Positive x, Negative y ( $+ , -$ ).
Example Calculations: * Point $S(3, -5)$ is in Quadrant IV (x = 3, y = -5). * Point $Q(-5, 3)$ is in Quadrant II (x = -5, y = 3).

Distance Between Two Points in the 2-D Plane

Axial or Parallel Segments: If two points have the same x or y value, distance is the absolute difference between the distinct coordinates. * Distance between $(x_1, y)$ and $(x_2, y)$ is $|x_2 - x_1|$ * Distance between $(x, y_1)$ and $(x, y_2)$ is $|y_2 - y_1|$
General Distance Formula: Derived from the Baudhāyana–Pythagoras Theorem. The distance between $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Example Calculation (Fig 1.7): * Point $A(3, 4)$ , Point $D(7, 1)$ . * Distance along x-axis ( $CD$ ): $7 - 3 = 4$ * Distance along y-axis ( $AC$ ): $4 - 1 = 3$ * Calculation: $AD = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\,units$
Effect of Reflection: Reflecting a shape (like triangle $ADM$ ) across an axis (e.g., the y-axis) changes the signs of specific coordinates but preserves the lengths of the sides. * Original $D(7, 1)$ reflects across y-axis to $D'(-7, 1)$ . * Original $A(3, 4)$ reflects across y-axis to $A'(-3, 4)$ . * $A'C' = |-3 - (-7)| = 4$ * $C'D' = |4 - 1| = 3$ * Resulting distance $AD = 5\,units$ .

Exercises and Discussion Topics

Accessibility Analysis: Questions regarding Reiaan’s room door $D_1R_1$ . * If $R_1$ is $(11.5, 0)$ and $D_1$ is $(x, y)$ , the width can be calculated. Standard door widths are evaluated based on whether a wheelchair ( $D_1R_1$ width) can fit through comfortably.
Bathroom Layout: Analysis of bathroom door ends $B_1(0, 1.5)$ and $B_2(0, 4)$ , showing a door width of $2.5\,units$ .
Coordinate Midpoints: Midpoint $M$ of segment $ST$ is found when coordinates are the average of the endpoints. * If $S(-3, 0)$ and $T(3, 0)$ , the midpoint is $M(0, 0)$ . * If $A(3, -4)$ and $M(-7, 1)$ is the midpoint, the endpoint $B(x, y)$ can be derived ( $x = -17, y = 6$ ).
City Model Application: * A city model uses two main roads (N-S and E-W) crossing at the center. * Streets are $200\,m$ apart. A scale of $1\,cm = 200\,m$ is used. * Example: Intersection $(2, 5)$ refers to the $2^{nd}$ N-S street and the $5^{th}$ E-W street. Only one unique intersection exists for $(4, 3)$ and $(3, 4)$ .
Computer Graphics: Screen coordinates (origin bottom-left). Screen is $800\,pixels$ wide and $600\,pixels$ high. * Circle $A$ : center $(100, 150)$ , radius $80$ . * Circle $B$ : center $(250, 230)$ , radius $100$ . * Calculations verify if circles exceed boundaries (x < 0, x > 800 or y < 0, y > 600) or intersect.

Chapter Summary

Positioning requires two perpendicular axes: horizontal (x-axis) and vertical (y-axis).
The point of intersection is the origin $(0, 0)$ .
The x-coordinate describes distance from the y-axis; the y-coordinate describes distance from the x-axis.
Coordinate signs by Quadrant: I ( $+, +$ ), II ( $(-, +$ ), III ( $-, -$ ), IV ( $+, -$ ).
Order matters: $(x, y) \neq (y, x)$ unless $x = y$ .
Distance formula for any two points $x_1, y_1$ and $x_2, y_2$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ , as per the Baudhāyana–Pythagoras Theorem.