Orienting Yourself: The Use of Coordinates

Historical Foundations of Coordinate Systems

• Origins of Grid-Based Thinking: The concept of using grids to define physical locations has deep roots in Bhārat. The first systematic urban-scale application occurred thousands of years ago in the Sindhu-Sarasvatĩ Civilisation.

  • Urban Planning: City streets were constructed with precision in North–South and East–West directions, spaced uniformly at intervals of approximately $10\,m$.

  • Practical Use: This served as a functional coordinate system; merchants located shops or warehouses by counting units of distance from the city centre.

• Baudhāyana (c. 800 BCE): Utilized North–South and East–West lines for advanced geometric constructions. His work developed the Baudhāyana–Pythagoras Theorem, laying the groundwork for coordinate geometry.

• Navigation and the Prime Meridian:

  • Ujjayinĩ: Described as early as the 4th century BCE in the early Siddhāntas as the central longitude meridian for world measurements.

  • Ptolemy (c. 150 BCE): Built on the work of Hipparchus to record latitudes and longitudes of thousands of locations, referring to Ujjayinĩ as ‘Ozine’.

• Mathematical Evolution:

  • Āryabhaᙩa (c. 499 CE): Replaced Greek ‘chords’ with ‘sines’ ($\sin(x)$), simplifying calculations for the coordinates of stars and cities. He mapped the sky using Celestial Coordinates measured from the ecliptic (the Sun's path).

  • Brahmagupta (c. 628 CE): Formalised zero ($0$) and negative numbers as algebraic entities. This made the modern four-quadrant Cartesian plane possible, as the origin is defined at zero and the negative axes represent values less than zero.

• Global Transmission:

  • Brahmagupta’s work was translated into Arabic as the Sindhind.

  • The Ujjayinĩ meridian was adopted into Arabic geography as ‘Arin’, serving as the zero-longitude reference.

  • Al-Bĩrũnĩ (c. 1000 CE): Studied the Siddhāntas and used Indian trigonometry to calculate Asian city coordinates. He perfected the ‘astrolabe’ for nautical navigation.

  • –mar Khayyām (c. 1100 CE): An expert in the Indian decimal system, he was the first to solve algebraic problems by interpreting them as geometry in a coordinate plane.

• European Formalisation:

  • Fermat (1636 CE) and Ren Descartes (1637 CE): Descartes formalised the definition that any point in a two-dimensional (2-D) plane can be represented by two numbers indicating distances from two perpendicular axes. This bridged algebra and geometry.

Conceptual Application: Reiaan’s Room

• Scenario: Reiaan, a student who is unable to see, moves to a new city. His sister Shalini, who has completed Grade 9, uses coordinate geometry to help him navigate their new home.

• The Tactical Grid: Shalini creates a physical map of the room's floor using a rectangular grid.

  • Materials: Pins to mark key points and thick wool/thread to connect points representing the corners of objects.

  • Scale: Shalini used a scale of $1\,cm : 1\,foot$.

• Limitations: Because the map is a 2-D floor plan, vertical elements (like the height of windows or the height of a table) cannot be represented.

The 2-D Cartesian Coordinate System Components

• Definition: A coordinate system is a structured framework (like grid lines) that uses numbers to describe exact physical locations.

• The Axes:

  • x-axis: The horizontal line of the system.

  • y-axis: The vertical line of the system.

  • Coordinate Axes: The collective name for both the x-axis and y-axis.

• The Origin (O): The point where the x-axis and y-axis intersect. Its coordinates are always $(0, 0)$.

• Coordinates of a Point: Represented as $(x, y)$.

  • x-coordinate: The perpendicular distance of a point from the y-axis, measured along the x-axis.

  • y-coordinate: The perpendicular distance of a point from the x-axis, measured along the y-axis.

• Standard Notation: While points can be written as $P = (x, y)$, it is common practice to drop the equals sign and write it as $P(x, y)$.

• Positional Rules:

  • Right of Origin: Positive x.

  • Left of Origin: Negative x (x < 0).

  • Upward from Origin: Positive y.

  • Downward from Origin: Negative y (y < 0).

• Points on Axes:

  • A point on the x-axis has coordinates of the form $(x, 0)$.

  • A point on the y-axis has coordinates of the form $(0, y)$.

The Cartesian Plane and Quadrants

• The Cartesian Plane: Also known as the coordinate plane or the xy-plane.

• Quadrants: The axes divide the plane into four distinct parts, numbered I through IV:

  • Quadrant I: Both x and y are positive ($+, +$).

  • Quadrant II: Negative x, positive y ($-, +$).

  • Quadrant III: Both x and y are negative ($-, -$).

  • Quadrant IV: Positive x, negative y ($+, -$).

• Identity Property: If $x \neq y$, then $(x, y) \neq (y, x)$. The points $(x, y)$ and $(y, x)$ only coincide if and only if $x = y$.

Distance Between Points in a 2-D Plane

• Parallel to Axes:

  • The distance between two points $(x_1, y)$ and $(x_2, y)$ on a line parallel to the x-axis is $|x_2 - x_1|$.

  • The distance between two points $(x, y_1)$ and $(x, y_2)$ on a line parallel to the y-axis is $|y_2 - y_1|$.

• General Distance Formula: For any two points $A(x_1, y_1)$ and $B(x_2, y_2)$, the distance is calculated using the Baudhāyana–Pythagoras Theorem:

  • Step 1: Find the horizontal shift $|x_2 - x_1|$.

  • Step 2: Find the vertical shift $|y_2 - y_1|$.

  • Step 3: Apply the theorem: $Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

• Case Study: Triangle ADM:

  • Points: $A(3, 4)$, $D(7, 1)$, and $M(9, 6)$.

  • Calculation for $AD$:

  • Distance along x-axis: $7 - 3 = 4$.

  • Distance along y-axis: $4 - 1 = 3$.

  • $AD = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\,units$.

  • Calculation for $DM$:

  • $DM = \sqrt{(9-7)^2 + (6-1)^2} = \sqrt{2^2 + 5^2} = \sqrt{29}\,units$.

  • Calculation for $MA$:

  • $MA = \sqrt{(9-3)^2 + (6-4)^2} = \sqrt{6^2 + 2^2} = \sqrt{40}\,units$.

Geometric Transformations and Reflections

• Reflection in the y-axis: When a shape (like triangle $ADM$) is reflected across the y-axis, the x-coordinates change sign while the y-coordinates remain the same.

  • Image of $A(3, 4) \rightarrow A'(-3, 4)$.

  • Image of $D(7, 1) \rightarrow D'(-7, 1)$.

  • Image of $M(9, 6) \rightarrow M'(-9, 6)$.

• Invariance: Reflection preserves the lengths of the sides of the triangle. For example, distance $A'D'$ remains $5\,units$, just like $AD$.

  • $A'C' = 4 - 1 = 3$.

  • $C'D' = -3 - (-7) = 4$.

  • $A'D' = \sqrt{3^2 + 4^2} = 5\,units$.

Practical Scenarios and Accessibility

• Door Width and Accessibility: In the exercise involving Reiaan’s room, the door $D_1R_1$ is discussed.

  • If $D_1$ is at $x = 10.5$ and $R_1$ is at $(11.5, 0)$, the door width is $1.0\,unit$ (which is $1\,foot$ in Shalini's scale).

  • Accessibility Question: A width of $1\,foot$ ($12\,inches$) is not a comfortable width for a room door and would not allow a person in a wheelchair to enter easily.

• Computer Graphics Application:

  • A screen is defined as a grid of pixels (e.g., $800\,pixels$ width $\times$ $600\,pixels$ height).

  • The origin $(0, 0)$ is at the bottom-left corner.

  • Objects like icons are placed using coordinate centers and radii.

  • To check if icons intersect, the distance between their center points $A(x_1, y_1)$ and $B(x_2, y_2)$ must be compared to the sum of their radii ($r_1 + r_2$).

Questions & Discussion

• Questions from Exercise 1.1:

  • Q: How far is the door $D_1R_1$ from the left wall (y-axis)?

  • A: The distance is represented by the x-coordinate of the door's start point.

  • Q: Is the bathroom door narrower or wider than the room door if bathroom endpoints are $B_1(0, 1.5)$ and $B_2(0, 4)$?

  • A: The width is $|4 - 1.5| = 2.5\,feet$. Compared to the room door (assuming $11.5 - 10.5 = 1\,foot$), the bathroom door is wider.

• Think and Reflect:

  • Q: What is the x-coordinate of a point on the y-axis?

  • A: The x-coordinate is always $0$.

  • Q: If $x \neq y$, then $(x, y) \neq (y, x)$. Is this claim true?

  • A: Yes. The order of coordinates matters in a 2-D plane.

  • Q: Are doors in schools suitable for wheelchairs?

  • A: This requires students to observe standard widths, which usually need to be at least $32–36\,inches$ for accessibility, whereas the example door was only $1\,foot$.

• Advanced Coordinate Challenges:

  • Collinearity: To check if points $M(-3, -4)$, $A(0, 0)$, and $G(6, 8)$ lie on the same straight line, one can compare the ratios of distance shifts or use the distance formula to see if $MA + AG = MG$.

  • Midpoint Identification: M is the midpoint of ST if its coordinates are the averages of the coordinates of S and T: $M\left(\frac{x_S+x_T}{2}, \frac{y_S+y_T}{2}\right)$.