GELE Mathematics — Analytic GeometryStudy Notes

The foundation of analytic geometry is the placement of geometric objects on a coordinate plane. A point P(x, y) is defined by its horizontal (x) and vertical (y) coordinates. Understanding how to measure distances and locate midpoints between points is crucial for surveying applications and structural layout in civil engineering. **Distance Formula Derivation:** Consider two points A(x₁, y₁) and B(x₂, y₂). By the Pythagorean theorem, the distance d between them is: d = √[(x₂ - x₁)² + (y₂ - y₁)²] This formula is used constantly in surveying to calculate distances between survey stations, and in structural engineering to determine member lengths in coordinate-based design. **Midpoint Formula:** The midpoint M of a segment AB is: M = ((x₁ + x₂)/2, (y₁ + y₂)/2) This is useful in civil engineering for finding the center of a plot of land or the midpoint of a structural member when performing symmetry analyses. **Practical Engineering Application:** In land surveying, if a plot corner at (100, 200) m is to be connected to a corner at (400, 500) m (coordinates relative to a site datum), the distance calculation ensures accurate boundary marking. Similarly, when designing a symmetric building about a centerline, the midpoint formula locates the geometric center.

In civil engineering, lines represent alignments—street centerlines, property boundaries, structural axes, and survey traverse lines. Understanding slope, parallel and perpendicular relationships, and the various forms of line equations is essential for layout and design. **Slope (Gradient):** The slope m of a line through points (x₁, y₁) and (x₂, y₂) is: m = (y₂ - y₁)/(x₂ - x₁) = Δy/Δx Slope represents the steepness of a line. In civil engineering, slope also refers to the vertical rise per horizontal unit distance (e.g., a 5% grade means 5 m rise per 100 m horizontal). **Relationship Between Slopes:** - Parallel lines have equal slopes: m₁ = m₂ - Perpendicular lines have slopes whose product is –1: m₁ × m₂ = –1 (negative reciprocal) These relationships are critical when designing perpendicular structures, setting up coordinate systems, or verifying that survey measurements are consistent. **Line Equations:** A line can be expressed in several forms: 1. Point-Slope Form: y – y₁ = m(x – x₁) (Used when you know a point and slope) 2. Slope-Intercept Form: y = mx + b (Used when you know slope and y-intercept) 3. General Form: Ax + By + C = 0 (Standard form used in distance and intersection calculations) 4. Two-Point Form: (y – y₁)/(y₂ – y₁) = (x – x₁)/(x₂ – x₁) (Used when you know two points) The general form Ax + By + C = 0 is particularly useful because it applies to all lines (including vertical lines, which have undefined slope in slope-intercept form).

In civil engineering, calculating the perpendicular distance from a point to a line is essential for: - Determining the shortest distance from a structure to a property boundary - Checking clearance distances from a survey line - Finding the offset distance in road design - Calculating the perpendicular distance from a column to a reference axis **Point-to-Line Distance Formula:** The perpendicular distance d from a point P(x₀, y₀) to a line Ax + By + C = 0 is: d = |Ax₀ + By₀ + C| / √(A² + B²) The absolute value ensures distance is always positive. The denominator √(A² + B²) is the normalization factor that accounts for the scaling of coefficients A and B. **Derivation Insight:** The distance formula is derived from the fact that the perpendicular from a point to a line is the shortest path. The numerator |Ax₀ + By₀ + C| represents the signed distance (before normalization), and the denominator normalizes it to the actual perpendicular distance. **Important Note:** When using this formula, the line must be in general form Ax + By + C = 0. If given in another form, convert it first.

The circle is the simplest conic section and is fundamental in civil engineering for: - Determining circular curves in highway and railway design (NSCP 2015) - Designing roundabouts and traffic circles - Finding areas of influence around structures - CAD applications and geometric layouts **Standard Form of a Circle:** (x - h)² + (y - k)² = r² Where: - (h, k) = center of the circle - r = radius - The circle consists of all points equidistant (distance r) from the center **General Form of a Circle:** Expanding the standard form gives: x² + y² + Dx + Ey + F = 0 Where D = -2h, E = -2k, F = h² + k² - r² **Converting From General to Standard Form:** Complete the square for both x and y terms: 1. Group x and y terms: (x² + Dx) + (y² + Ey) + F = 0 2. Complete the square: (x + D/2)² - (D/2)² + (y + E/2)² - (E/2)² + F = 0 3. Simplify: (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F Compare with (x - h)² + (y - k)² = r² to identify center and radius. **Key Properties:** - A circle passes through any point (x, y) satisfying the equation - The distance from the center to any point on the circle is always r - A circle has no eccentricity (eccentricity e = 0) - Useful for determining catchment areas, sight distances, and clearance zones

The ellipse appears in civil engineering for: - Highway transition curves (spiral to circular curve connections) - Building floor plans (elliptical arches and domes) - Structural analysis involving eccentric loading - Surveying and coordinate transformations **Standard Form of an Ellipse (Horizontal Major Axis):** (x - h)²/a² + (y - k)²/b² = 1 Where: - (h, k) = center of the ellipse - a = semi-major axis (longer radius, along x-axis) - b = semi-minor axis (shorter radius, along y-axis) - a > b (for horizontal orientation) **For Vertical Major Axis:** (x - h)²/b² + (y - k)²/a² = 1 (Note: a and b swap positions; a > b still) **Key Relationships:** - Foci are located at distance c from the center, where c² = a² - b² - Eccentricity: e = c/a (always 0 < e < 1 for an ellipse) - The sum of distances from any point on the ellipse to the two foci is constant = 2a - When a = b, the ellipse becomes a circle (special case, e = 0) **Geometric Properties:** - An ellipse is symmetric about both its axes - The major axis passes through both foci - The minor axis is perpendicular to the major axis - Ellipses are used in highway design as transition curves (though parabolas are more common in modern NSCP 2015 designs)

The parabola is the most important conic in civil engineering applications: - Highway vertical alignment curves (profile grades, NSCP 2015) - Arch structures and suspension bridges (cables follow parabolic paths) - Drainage swales and surface water flow paths - Satellite dish and reflector antennas - Trajectory of projectiles in ballistics **Standard Forms of a Parabola:** Vertex at (h, k), opening to the right: (y - k)² = 4a(x - h) Focus: (h + a, k) Directrix: x = h - a Vertex at (h, k), opening upward: (x - h)² = 4a(y - k) Focus: (h, k + a) Directrix: y = k - a Where: - a = focal distance (distance from vertex to focus) - The parabola opens toward the focus - The directrix is on the opposite side of the vertex from the focus - Distance from any point on the parabola to focus = distance to directrix (defining property) **Key Properties:** - Vertex (h, k) is the turning point - Axis of symmetry passes through the vertex and focus - Eccentricity e = 1 (all parabolas have the same eccentricity) - If a > 0, parabola opens right (or up); if a < 0, opens left (or down) - The focal parameter 4a is the coefficient in the equation **Highway Vertical Alignment (NSCP 2015):** Vertical parabolic curves are standard for grade transitions. The equation (x - h)² = 4a(y - k) describes a vertical alignment where: - PVC (Point of Vertical Curve) = vertex - x represents station distance along the horizontal - y represents elevation - The parameter a determines curve length

The hyperbola has specialized applications in civil engineering: - Radar and navigation systems (LORAN positioning—two transmitters locate by hyperbolic curves) - Trajectory analysis under certain field conditions - Acoustic and electromagnetic wave propagation - Cooling towers (hyperboloid surfaces) **Standard Forms of a Hyperbola:** Opening Left-Right (Horizontal Transverse Axis): (x - h)²/a² - (y - k)²/b² = 1 Foci: (h ± c, k), where c² = a² + b² Asymptotes: y - k = ±(b/a)(x - h) Opening Up-Down (Vertical Transverse Axis): (y - k)²/a² - (x - h)²/b² = 1 Foci: (h, k ± c), where c² = a² + b² Asymptotes: y - k = ±(a/b)(x - h) Where: - (h, k) = center - a = distance from center to vertex along transverse axis - b = distance from center to point on asymptote perpendicular to transverse axis - c = distance from center to focus (always c > a for hyperbola) - The hyperbola has two separate branches - Vertices are at distance a from the center **Key Properties:** - Eccentricity: e = c/a > 1 (always greater than 1 for hyperbola) - The difference of distances from any point to the two foci is constant = 2a - Asymptotes define the "envelope" of the hyperbola as it extends to infinity - Unlike the ellipse (+), the hyperbola uses subtraction (−) - As e → 1⁺, the hyperbola branches become narrow; as e increases, they widen

The four conic sections are classified by their eccentricity (e), a parameter that measures how much the curve deviates from a circle. Understanding which type of conic you're dealing with is essential for solving complex engineering problems. **Eccentricity Classification:** 1. **Circle: e = 0** - Equation: (x - h)² + (y - k)² = r² - All points are equidistant from a single center - Used for roundabouts, circular curves, areas of influence 2. **Ellipse: 0 < e < 1** - Equation: (x - h)²/a² + (y - k)²/b² = 1 (a ≠ b) - Two foci; sum of distances to foci is constant - c² = a² - b² - Used for transition curves, architectural elements 3. **Parabola: e = 1** - Equation: (y - k)² = 4a(x - h) or (x - h)² = 4a(y - k) - One focus and directrix; distances are equal - Most important for highway vertical alignment (NSCP 2015) - Used for arches, suspensions, drainage paths 4. **Hyperbola: e > 1** - Equation: (x - h)²/a² - (y - k)²/b² = 1 (or with y first) - Two foci; difference of distances to foci is constant - c² = a² + b² - Used for navigation systems, structural analysis **Quick Identification From Equation:** - If x² and y² have the same sign and same coefficient → **Circle** - If x² and y² have the same sign but different coefficients → **Ellipse** - If x² and y² have opposite signs (one + and one −) → **Hyperbola** - If only one squared term (x² or y², not both) → **Parabola** **Relationship Between Ellipse and Hyperbola:** - Ellipse: c² = a² - b² (subtract) - Hyperbola: c² = a² + b² (add) - This is the key difference in calculating focal distance