![]() ![]() The SRI constant has no effect on this calculation. The result for a parabolic arc length is not iterative, it is exact. The expected accuracy of a typical arc length calculation for an hyperbola ( x 31, a 20, p 7.2) dependent upon ‘SRI’ is shown below: X = the horizontal distance along the x-axis to point ' P' on the curve a = the horizontal aspect dimension used in the definition of the curve’s eccentricity ( e) and its parameter ( p) b = the vertical aspect dimension used in the definition of the curve’s eccentricity ( e) and its parameter ( p) : ℓ = 25.55755004Ĭompared with a more accurate value of 25.527039 The ParabolaĪll properties are calculated for any point (x,y) on the positive quarter of a parabola bounded by co-ordinate '0,0' and positive values for ' x' and ' y' (see Fig 3) Input Data Ellipses will display a warning if you enter a smaller value for ‘ x’ than for ‘ a’. This calculation only works if ‘ x’ is greater than ‘ a’. ![]() The HyperbolaĪll properties are calculated for any point (x,y) on the positive quarter of an hyperbola bounded by co-ordinate ' a,0' and positive values for ' x' and ' y' (see Fig 4) Ellipses will display a warning if you enter a value for ‘ a’ less than ‘ b’. the ellipse is always flattened vertically. This calculation only works if ‘ a’ (x-axis) is greater than ‘ b’ (y-axis), i.e. The value of ‘ x’ must be greater than or equal to ‘0’ and less than ‘ a’ Construction Drawings of a Conical EllipseĪll properties are calculated for any point (x,y) on the quarter of an ellipse bounded by centre co-ordinate '0,0' and positive values for ' x' and ' y' (see Fig 5) The mathematical properties and relationships of these elliptical curves are defined below: Circleįig 7. The distance from the focus to the directrix of all the elliptical curves = p/ e ![]() Half the width of the curve at the focus ( F or F₁ or F₂) is called its parameter ( p) or ordinate and is dependent upon the angle (or slope) of the cone. ![]() the ratio of the distances from any point ( P) on the curve to its focus ( F) and its directrix circle: R: r=0 ( Fig 2) parabola: r: r=1 ( Fig 3) hyperbola: r₁: r>1 ( Fig 4) ellipse: r₁: r<1 ( Fig 5) The shape of each conic is defined by its eccentricity ( e), i.e. N = d²+e²-4.f.(a+c) n0: two parallel straight lines which can be deduced from the following table: The determinant (δ) for the above equation is calculated thus You can solve the above equation using determinants, Where a, b, c, d, e & f are the variables that define the type of curve (circle, ellipse, parabola or hyperbola) Note: For all circles a = 1, b = 0, c = 1 the circle, the ellipse, the parabola and the hyperbola) is called a conic because it is defined by the angle it cuts through a circular cone (or nappe).Įvery conic (curve) is defined by the same equation: Elliptical Family of CurvesĮach curve in the family of ellipses ( Fig 1 i.e. The best known practical example of an ellipse is Johannes Kepler’s law for planetary orbits, the size and shape (eccentricity) of which are defined by the gravitational attraction between a planet and its sun. ![]()
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