History of Analytic Geometry PDF

HISTORY OF ANALYTIC GEOMETRY Analytic Geometry Analytic geometry is the algebraic study of geometry using a coordinate plane. Employing the concept of the Cartesian coordinate system, it established the relationship between geometry and algebra making way for the development of mathematics. It...

HISTORY OF ANALYTIC GEOMETRY Analytic Geometry Analytic geometry is the algebraic study of geometry using a coordinate plane. Employing the concept of the Cartesian coordinate system, it established the relationship between geometry and algebra making way for the development of mathematics. It provides the identity of points, lines, and curves by defining their relationship through algebraic equations, describing measurements and locations of geometric figures, and providing proofs to certain existing theorems. Thus, analytic geometry is useful in both Euclidian and non-Euclidian geometry. The application of analytic geometry with the aid of a Cartesian coordinate system starts with giving a unique identity to each point on a cartesian plane. The cartesian plane consists of two perpendicular lines called the axes; one is horizontal or the x- axis, and the other is vertical or the y-axis. The point at which these axes intersect at (0, 0), is called the origin. It is where the horizontal and vertical distances of a point are referenced. The horizontal distance from the origin is the x-coordinate while the vertical distance from the origin is the y-coordinate, collectively known as the coordinates of the point, represented by an ordered pair (x, y). The cartesian plane is divided into four quadrants. A point located to the right and above the origin has both positive values of x and y coordinates and can be found on the 1st Quadrant. A point to the left and above the origin has a negative x-coordinate and positive y-coordinate and can be found on the 2nd Quadrant. A point to the left and below of the origin has both negative values of x and y coordinates, and can be found on the 3rd Quadrant. A point to the right and below the origin has a positive x-coordinate and negative y-coordinate, and can be found on the 4th Quadrant. The x-coordinate is also called the abscissa while the y-coordinate is called the ordinate. Ancient Foundations (c. 250 BCE - 1st Century CE) Euclid (c. 300 BCE): His Elements formalized many geometric principles but lacked algebraic representation. Apollonius of Perga (c. 262–190 BCE): Known as the "Great Geometer," he studied conic sections (ellipse, parabola, hyperbola) and derived properties of these curves, foreshadowing analytic geometry Islamic Golden Age (8th–14th Century) Omar Khayyam (1048–1131 CE): Solved cubic equations using geometric methods involving conic sections. Al-Khwarizmi: Advanced algebra, laying the foundation for later connections between algebra and geometry. Renaissance & Early Algebra (16th Century) François Viète (1540–1603): Introduced symbolic algebra, using letters for variables and constants, enabling the connection between algebra and geometry. The Birth of Analytic Geometry (17th Century) René Descartes (1596–1650): His work La Géométrie (1637) introduced the Cartesian coordinate system, marking the formal beginning of analytic geometry. He used algebra to study curves and solved geometric problems through equations. Pierre de Fermat (1601–1665): Independently developed similar methods, analyzing conic sections and establishing algebraic approaches to geometry. John Wallis (1616–1703): Introduced negative coordinates and expanded Descartes’ work on conic sections. 19th Century: Vector Analysis & Higher Dimensions William Rowan Hamilton (1805–1865): Introduced quaternions, extending coordinate geometry to four dimensions. Julius Plücker (1801–1868): Unified analytic and projective geometry through homogeneous coordinates. 20th Century & Modern Applications Computer Graphics and CAD: Analytic geometry is essential in 3D modeling, video games, and engineering. Robotics & Machine Learning: Used in motion planning, data visualization, and high-dimensional projections.  Key Concepts in Analytic Geometry 1. Vectors: Quantities with magnitude and direction, analyzed in 2D or 3D coordinate systems. 2. Projections: Representation of figures on a surface, used in applications like perspective drawings. 3. Conics: Geometric shapes derived from intersecting a plane with a cone, including: Circle Ellipse Parabola Hyperbola Each conic is represented by a unique algebraic equation.  FORMULAS FOR ANALYTIC GEOMETRY The main formulas used in analytic geometry give appropriate importance to the properties of geometric figures. Among these are the distance formula, midpoint formula, and slope formula. Other formulas in analytic geometry can be derived with the aid of these formulas. Distance formula – This is useful in defining the separation of two given points. It can also be used in identifying the location of a point given the coordinate of the other point along with the distance between them. The distance can be solved with this formula: The midpoint formula is used to find the midpoint between two points whose coordinates are known to us. The midpoint formula is also used to find the coordinates of the endpoint if we know the coordinates of the other endpoint and the midpoint. In the coordinate plane, if a line is drawn to connect two points (4, 2), and (8, 6), then the coordinates of the midpoint of the line joining these two points are ({4 + 8}/2, {2 + 6}/2) = (12/2, 8/2) = (6, 4). Midpoint refers to a point that is exactly in the middle of the line segment joining two points. The two reference points are the endpoints of a line segment, and the midpoint is lying in between the two points. The midpoint divides the line joining these two points into two equal halves. Further, if a line is drawn to bisect a line segment joining these two points, the line passes through the midpoint. Example: Using the midpoint formula, find the midpoint between points X(5, 3) and Y(7, 8). Solution: Let M be the midpoint between X and Y. M = ((5 + 7)/2, (3 + 8)/2) = (6, 11/2) Therefore, the coordinates of the midpoint between X and Y is (6, 11/2). Slope Formula is used to calculate the inclination or steepness of a line. It finds application in determining the slope of any line by finding the ratio of the change in the y-axis to the change in the x-axis. The slope of a line is defined as the change in the “y” coordinate with respect to the change in the “x” coordinate of that line. Examples Using Slope Formula Example 1: Find the slope of a line whose coordinates are (2,9) and (4,1)? Solution: To find: The slope of the line with coordinates (2,9) and (4,1) Given, (x1, y1) = (2, 9) and (x2, y2) = (4, 1) The slope formula is m = (y2 – y1)/(x2 – x1) m = (1 − 9)/(4 − 2) m = -8/2 = -4 Example 2: Determine the value of b, if the slope of a line passing through the points (b, 7) and (8, -5) is 6. Solution: To find: the value of b Given, Slope = m = 6, Points: (x1, y1) = (b, 7) and (x2, y2) = (8, -5) We know that Slope (m) = (y2 – y1)/(x2 – x1) 6 = (-5-7)/(8-b) 6 = (-12)/(8-b) -2= (8-b) -2-8 = -b, b = 10 Answer: The value of b = 10. THANK YOU!