2D Geometry Fundamentals

• Notes on 2D Geometry

Fundamental Concepts

• Covers points, straight lines, circles, parabolas, ellipses, and hyperbolas

Point

• A point is defined by its coordinates
• Cartesian Coordinates: defined by x and y axes.
• Polar Coordinates: defined by radius (r) and angle (theta)
• Relation between Cartesian and Polar Coordinates: x = rcos(θ), y = rsin(θ), r² = x² + y², θ = tan⁻¹(y/x)
• Image Point Reflections:
• With respect to x-axis:(x,-y)
• With respect to y-axis:(-x,y)
• With respect to the line x = y:(y,x)
• With respect to the origin:(−x,−y)
• Distance Formula: The distance between two points P(x₁, y₁) and Q(x₂, y₂) is given by PQ = √((x₂ − x₁)² + (y₂ − y₁)²)

Straight lines

• A general equation of a straight line is required of the form "ax + by + c = 0"
• Equations Summary
• x-axis: y = 0
• y-axis: x = 0
• parallel to the x-axis: y=a
• parallel to the y-axis: x=a
• Combined equation of axes is given by xy = 0
• Slope (m) = tan θ
• Slope Formula: m = (y₂ - y₁) / (x₂ - x₁) = Δy/Δx.
• For acute angles, the slope is positive; for obtuse angles, the slope is negative.
• Slope-intercept form: y = mx + c, where m is the slope and c is the y-intercept
• Condition for concurrency of lines: Three lines are concurrent if a determinant involving their coefficients is zero

Division of a line

• If a point (x, y) divides a line between points P(x₁, y₁) and Q(x₂, y₂) in the ratio m₁:m₂.
• For internal division the coordinates of M = ((m₁x₂ + m₂x₁) / (m₁ + m₂), (m₁y₂ + m₂y₁) / (m₁ + m₂))
• For external division M = ((m₁x₂ - m₂x₁) / (m₁ - m₂), (m₁y₂ - m₂y₁) / (m₁ - m₂)).
• Division in ratio with axes
• Ratio using x-axis: −y₁/y₂
• Ratio using y-axis: −x₁/x₂

Properties Related to Multiple Lines

• For four sides with vertices A, B, C, & D:
• If AB = BC = CD = DA and AC = BD, then it's a square
• If AB = BC = CD = DA and AC ≠ BD, then it's a rhombus
• If AB = CD and BC = AD and AC = BD, then it's a rectangle
• If AB = CD and BC = AD and AC ≠ BD, then it's a parallelogram
• If diagonal slopes are perpendicular, it is a square or rhombus
• If diagonal slopes are not perpendicular, it is a rectangle or parallelogram

Triangle

• Centroid Coordinates: ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
• Area of a Triangle: area = (1/2) * determinant of x₁, y₁, 1 ; x₂, y₂, 1 ; and x₃, y₃, 1
• Area When Equations of Sides are Given. area = (1/(2C₁C₂C₃)) * determinant squared of a₁, b₁, c₁ ; a₂, b₂, c₂; and a₃, b₃, c₃

Circle

• General Equation: x² + y² + 2gx + 2fy + c = 0
• Center: (−g, −f)
• Radius:(r) = √(g² + f² − c)
• Diameter form: (x − x₁) (x − x₂) + (y − y₁) (y − y₂) = 0
• Intersection points in circle
• (g² + f² − c) > 0, Circle intersects x- axis at 2 points
• (g² + f² − c) = 0, x- axis touches
• (f² + f² − c) > 0, Circle intersects y- axis at 2 points
• Equation for touching x-axis:(x−h)² + (y−a)² = a² xy for touch 5(y"
• Position is described based on c1c2 where c is center and r is radius
• Condition in terms of formula 5 and point (X,Y)
• Length of Tangent: √[x₁² + y₁² + 2gx₁ + 2fy₁ + c ]

Circle’s properties

• Condition for it to be a circle if a = 1, h = 0
• length of the intercepts of circle on axis has formulas
• X’ Axis 2 √ {g²-c}
• Y’ Axis 2 √ { f ²-c}
• Tangency Relation with Line
• line y- mx+ c has the relation of value being the d =
• y- mx+ c has value and d be length value by |4*
• S=4 5(y"
• y- mx+ c S=c=
• If Circle has value, touching x-axis
• touching x 5(y and y as touching
• Diameter for circle has the functions on which value is created along radius
• Diameter Function is created against function in circle

Parabola

• General equation for condition of parabola is a function.
• Defined by focus,vertex, axis and directex
• Defined 4. Given by y"=- 4 ax and equation value of 5
• General formulas and conditions defined

Conic Sections - Ellipse

• General formula 4 = ax" + 2 hrxy + by+ 2gx + 2fy +C=0 is followed.
• e5 for it to be valid as conic sections
• Where standard forms may vary, with focus having direct relations.
• 41 for different formatives based on conditions