The cubic parabola is **a simple function of the form of y = f(x) and is based on the acknowledgment that its length is equal to its projection on axis X**. Clothoid is a transition curve in the form of x = f(l), y = f(l), having as main characteristic the linearity of curvature variation versus its length.

The cubic parabola function is y=kx3 (1) The “main” elements in railway transition curve are: The radius of curvature at the end of transition, the length L of the curve, the length l of its projection on x axis and the coefficient k.

Their graphs are parabolas. To find the x-intercepts we have to solve a quadratic equation. The vertex of a parabola is a maximum of minimum of the function. Then four points not in a line nor in a parabola determine a cubic function.

Railways are primarily constructed with straight lines and circular arcs, a transition curve makes a smooth transition between two curves with different curvature. This use gave the curve the name of cubic spiral. The spiral of the The American Railway Engineering Association, the A.R.E.A.

Examples include the cissoid of Diocles, conchoid of de Sluze, folium of Descartes, Maclaurin trisectrix, Maltese cross curve, right strophoid, semicubical parabola, serpentine curve, Tschirnhausen cubic, and witch of Agnesi, as well as elliptic curves such as the Mordell curve and Ochoa curve.

In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs.

Sketching Cubics

- Find the x-intercepts by putting y = 0.
- Find the y-intercept by putting x = 0.
- Plot the points above to sketch the cubic curve. e.g. Sketch the graph of y = (x − 2)(x + 3)(x − 1)
- Find the x-intercepts by putting y = 0. ...
- Find the y-intercepts by putting x = 0. ...
- Plot the points and sketch the curve.

A spiral curve can be used to provide a gradual transition between tangent sections and circular curves. While a circular curve has a radius that is constant, a spiral curve has a radius that varies along its length.

Lemniscate is a type of transition curve which is used when the deflection angle is very large. In lemniscate the radius of curve is more if the length of chord is less.

*No Quiz* Construction Surveying Curves COMPOUND CURVES. A compound curve is two or more simple curves which have different centers, bend in the same direction, lie on the same side of their common tangent, and connect to form a continuous arc.

A cubic function has the standard form of f(x) = ax^{3} + bx^{2} + cx + d. The "basic" cubic function is f(x) = x^{3}. You can see it in the graph below. In a cubic function, the highest power over the x variable(s) is 3.

Cubic graphs are curved but can have more than one change of direction.

y = x^3/6RL is equation of cubic parabola.

Remember the pattern for parabolas: vertical: y=a(x-h)^{2}+k horizonal: x=a(y-k)^{2}+h. Often parabolas are already listed in this format, but sometimes they are not. In this case, you must put them into the graphing format by completing the square.

Standard Equation of Parabola

The simplest equation of a parabola is y^{2} = x when the directrix is parallel to the y-axis. In general, if the directrix is parallel to the y-axis in the standard equation of a parabola is given as: y^{2} = 4ax.

Spiral curves are generally used to provide a gradual change in curvature from a straight section of road to a curved section. They assist the driver by providing a natural path to follow. Spiral curves also improve the appearance of circular curves by reducing the break in alignment perceived by drivers.

Summit curves are vertical curves with gradient upwards. Summit curve. Summit curves are vertical curves with gradient upwards. They are formed when two gradients meet as illus-trated in any of the following four ways: = when a positive gradient meets another positive gradient.

PC = Point of Curve. PT = Point of Tangent.

A track transition curve, or spiral easement, is a mathematically-calculated curve on a section of highway, or railroad track, in which a straight section changes into a curve. It is designed to prevent sudden changes in lateral (or centripetal) acceleration.

Spirals. A spiral is a curved pattern that focuses on a center point and a series of circular shapes that revolve around it. Examples of spirals are pine cones, pineapples, hurricanes.

helix. a 3-dimensional spiral. Rhumb line (also loxodrome) type of spiral drawn on a sphere.

A cubic equation is an algebraic equation of third-degree.

The general form of a cubic function is: f (x) = ax^{3} + bx^{2} + cx^{1} + d. And the cubic equation has the form of ax^{3} + bx^{2} + cx + d = 0, where a, b and c are the coefficients and d is the constant.

A cubic function is a polynomial function of degree 3. So the graph of a cubic function may have a maximum of 3 roots. i.e., it may intersect the x-axis at a maximum of 3 points. Since complex roots always occur in pairs, a cubic function always has either 1 or 3 real zeros. It cannot have 2 real zeros.

In geometry, cubic units can be defined as the units used to measure volume. The volume of a unit cube whose length, width and height are 1 unit each is 1 cubic unit. Here, the rectangular prism is made up of smaller unit cubes.