Orienting Yourself: The Use of Coordinates

Historical Foundations of Coordinate Geometry

• Definition of a Coordinate System: A system of coordinates is a structured framework (comparable to grid lines on a map or graph paper) used to describe the exact physical locations of points or objects using numbers.

• Ancient Roots in Bhārat:     * Sindhu-Sarasvatĩ Civilisation: The first systematic application of grids occurred thousands of years ago on an urban scale. City streets were constructed with striking precision in North–South and East–West directions at uniform distances of approximately $10\,m$ apart. This allowed merchants to find locations by counting units of distance from the city centre.     * Baudhāyana (c. 800 C.E.): Utilised East–West and North–South lines for geometric constructions. He developed the Baudhāyana–Pythagoras Theorem, laying the essential foundation for coordinate geometry.

• Evolution of Global Navigation and Mapping:     * Ujjayinĩ: Known as the central longitude meridian for the ancient world. It was documented as early as the 4th century BCE in the early Siddhāntas as the reference point for all other measurements.     * Ptolemy (c. 150 BCE): Building on the work of Hipparchus, he described the latitudes and longitudes of thousands of locations, including Ujjayinĩ (referred to as ‘Ozine’).     * Āryabhaᙩa (c. 499 CE): Revolutionised calculations by replacing Greek ‘chords’ with ‘sines’ ($\sin$). He mapped the sky using Celestial Coordinates, measuring distances from the ecliptic (the sun’s path).     * Brahmagupta (c. 628 CE): Formalised the concept and algebraic use of zero and negative numbers. This work was a prerequisite for the modern four-quadrant Cartesian plane, where the ‘origin’ is zero and negative axes represent values less than zero.

• Transmission and Formalisation:     * Arabic Transition: Brahmagupta’s work was translated into Arabic as the Sindhind. The Ujjayinĩ meridian was renamed ‘Arin’ and served as the zero-longitude reference for early Arabic maps.     * Al-Bĩrũnĩ (c. 1000 CE): Travelled to India, studied the Siddhāntas, and applied Indian trigonometric methods to calculate city coordinates across Asia. He also perfected the astrolabe for sailors.     * Ömar Khayyām (c. 1100 CE): An expert in the Indian decimal system and algebraic formalism; the first to solve algebraic problems using geometry by interpreting them as coordinates in a plane.     * European Formalisation: Following the work of Pierre de Fermat (1636 CE), René Descartes (1637 CE) formalised the 2-D plane. Every point can be defined by two numbers representing distances from two perpendicular axes, merging algebra and geometry.

The 2-D Cartesian Coordinate System

• Core Components:     * Coordinate Axes: Two lines positioned at right angles (perpendicular) to each other.     * x-axis: The horizontal coordinate line.     * y-axis: The vertical coordinate line.     * Origin (O): The point where the x-axis and y-axis intersect. Its coordinates are defined as $(0, 0)$.     * Coordinate Plane: Also known as the Cartesian plane or xy-plane.

• Sign Conventions:     * Positive Distances: Points located to the right of O (along the x-axis) or upwards from O (along the y-axis).     * Negative Distances: Points located to the left of O (along the x-axis) or downwards from O (along the y-axis).

• Point Representation:     * A point P is represented as $(x, y)$.     * x-coordinate: Represents the perpendicular distance of P from the y-axis, measured along the x-axis.     * y-coordinate: Represents the perpendicular distance of P from the x-axis, measured along the y-axis.     * Points on Axes: Any point on the x-axis has coordinates $(x, 0)$. Any point on the y-axis has coordinates $(0, y)$.

• Quadrants: The axes divide the plane into four segments:     1. Quadrant I: Positive x, positive y ($+, +$).     2. Quadrant II: Negative x, positive y ($-, +$).     3. Quadrant III: Negative x, negative y ($-, -$).     4. Quadrant IV: Positive x, negative y ($+, -$).

Distance and Geometry in the Plane

• Distance between Parallel Points:     * For two points on a horizontal line ($(x_1, y)$ and $(x_2, y)$), the distance is the absolute value of the difference in x: $|x_2 - x_1|$.     * For two points on a vertical line ($(x, y_1)$ and $(x, y_2)$), the distance is the absolute value of the difference in y: $|y_2 - y_1|$.

• General Distance Formula (Baudhāyana–Pythagoras Theorem):     * The distance between any two points $A(x_1, y_1)$ and $D(x_2, y_2)$ is found by constructing a right-angled triangle where the horizontal change is $(x_2 - x_1)$ and the vertical change is $(y_2 - y_1)$.     * Formula: $\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

• Length Preservation in Reflections:     * Reflecting a shape (like $\triangle ADM$) across an axis (e.g., the y-axis) changes the signs of the coordinates (e.g., $(x, y)$ becomes $(-x, y)$), but the lengths of the sides remain identical.     * Example: If $A(3, 4)$ and $D(7, 1)$, then $AD = 5$. Their reflected images $A'(-3, 4)$ and $D'(-7, 1)$ also yield $A'D' = 5$.

Practical Case Study: Reiaan’s Room

• Scenario Background: Reiaan, who is blind, moved to a new city. His sister Shalini used Coordinate Geometry to help him navigate his new environment.

• Tactile Map Creation:     * Materials: A rectangular grid with pins and threads.     * Scale: $1\,cm : 1\,foot$.     * Visual Constraints: The floor plan shows objects on the floor. Vertical features, like the height of windows, cannot be marked on a 2-D floor map.

• Accessibility Analysis (Exercise 1.1):     * Doorway Width: If the door starts at $(10, 0)$ (derived from D1) and ends at $R1(11.5, 0)$, the width is $1.5\,feet$.     * Wheelchair Access: A narrow width (like $1.5\,feet$) is typically insufficient for comfortable wheelchair access, which requires specific standard widths.     * Bathroom Door: Represented by $B1(0, 1.5)$ and $B2(0, 4)$, the width is $|4 - 1.5| = 2.5\,feet$. This is wider than the room door described.

Questions & Discussion

• Standard Door Widths: Questions were raised regarding standard widths in schools and homes to determine if they are suitable for persons in wheelchairs.
• Midpoint Identification (Question 9):     * S $( -3, 0)$, M $(0, 0)$, T $(3, 0)$: M is the midpoint because it is equidistant and collinear.     * S $(2, 3)$, M $(3, 4)$, T $(4, 5)$: M is the midpoint.     * S $(0, 0)$, M $(0, 5)$, T $(0, -10)$: M is not the midpoint; the distance SM is 5, but MT is 15.
• Circle Verification (Question 12):     * A point $(x, y)$ lies on a circle centered at origin $(0, 0)$ if the distance $\sqrt{x^2 + y^2}$ equals the radius $r$.     * For $A(1, -8)$, $B(-4, 7)$, and $C(-7, -4)$, the distance from origin for all is $\sqrt{65}$, meaning they lie on circle K with radius $\sqrt{65}$.     * $D(-5, 6)$ gives $\sqrt{(-5)^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61}$. Since $\sqrt{61} < \sqrt{65}$, point D is inside the circle.     * $E(0, 9)$ gives $\sqrt{0^2 + 9^2} = 9 = \sqrt{81}$. Since $\sqrt{81} > \sqrt{65}$, point E is outside the circle.
• City Model (Question 14):     * Scale: $1\,cm = 200\,m$.     * Intersections like $(4, 3)$ and $(3, 4)$ are distinct. There is only one unique intersection for the 4th N-S street and 3rd E-W street ($4, 3$) and one for the 3rd N-S and 4th E-W street ($3, 4$).