3. Horizontal Curve

Probably the most prevalent curve that surveyors deal with are horizontal which use circular arc sections. These are also second degree polynomials but unlike a parabola, they have constant radii. Another difference is there is really no dependent-independent variable distinction. The data points for a horizontal curve are coordinates (X/Y or N/E), neither direction superior to the other.

Being a second-degree polynomial, three points are needed to define it.

Equation I-9 is used to solve the arc if one of the points is the radius point, O, Figure I-12.

				Equation I-9
where: Np, Ep - coordinates of a point on the arc No, Eo - coordinates of the arc radius point
Figure I-12 Including radius point

Equations I-10 and I-11 are used to solve the arc if all three points are on the arc, Figure I-13.

		Equation I-10
		Equation I-11
Figure I-11 Only arc points

Four points allows a least squares solution.

Example

Determine the radius for the arc in Figure I-12 using the coordinates of the four arc points in Table I-4.

Figure I-12 Least Squares Arc Fitting

Use the coordinates and Equation I-10 to set up the observation equations. Include a residual on the constant term.

Create the initial matrices

Intermediate matrix

Solution matrix

Substitute the solution coefficients into the arc equation.

Determine the curve radius

Residuals

The observation equation form of equations I-9 and I-10 are:

	Equation I-12
	Equation I-13

In Equation I-12 the residual is on the radius which is a function of the coordinates.

In Equation I-13 the residual is on the arc point coordinates.

This makes the residuals perpendicular to the arc (radial), Figure I-13, since there is no dependent-independent variable condition.

Figure I-13 Curve Fit Residuals