**What is a Cross-sectional study?**

Simply put, this type of study is simply a comparison of two areas with x and y values. The x and y values will be in a horizontal and vertical position, respectively, at the same time. There are many types of crosses, and they come from measurements taken at many different angles. The types include:

**Quatrefoil** – This is an angle measurement, and involves two intersecting lines. These are set at various angles to each other, and the x value of one can be compared to the x value of the other. Parabolometrics and Parabola are two types of parabolic measurements. A parabolic plane will include an area that touches the surface of the area being studied. The Quatrefoil angle is often used to study area patterns on curved surfaces, for instance.

**Hyperbola** – This is a measure of the angle between two parabolic segments. It is often used to find parallels between areas that are otherwise parallel. Hyperbolas can also be used to find parallels along a surface when the parabola is oriented along the axis of that surface. In addition to parabolism, hyperbolas can also be studied for hyperbola transformations.

**Parabolic** Arc – This involves finding a parabolic parabola that forms an interval, or is a parabola in itself. The parabola may not coincide with a parabolism, however. For instance, if a point on the x axis is plotted along a parabolic chord, this will not be a parabolic arc. However, it is possible to plot a parabola on a parabolic chord, and this is what is often done in a parabola study.

**Cross Section** – A parabola can be plotted as a section through a plane or some other surface. A cross section of a parabola, therefore, just shows the portion of the parabola opposite the symmetry. This can be used in what is a cross-sectional study to find parallels between two points on the plane. For instance, if we plot the parabola below a paraboloid, this will show the section through a plane which goes through the symmetry.

Parabolic Projection – A parabola can also be plotted on a parabolic projection. For instance, if we plot the parabola tangent to a right-angle, this will show the area beneath the parabola. The conic section will then be found by plotting a parabolic projection onto the plane of the conic. One can also find parallels between parabolas on a cylindrical projection. However, for these types of projections, it is usually more convenient to use spherical projections.

**Angles, Shapes, and Surfaces** – The angle or change of angle at any point on a given surface can be used in what is a cross-sectional study. The main figure to remember is that any change of angle, if it is small, will be insignificant. It may not alter the parabola significantly, but it may alter the parabola’s axis of symmetry by one or more degrees. Surfaces can be chosen in many different ways. The simplest surface is the flat plane. If this is chosen, all that is needed to find the parabola’s parallels is to find the parallel on the surface.

Another way of choosing a surface is by choosing a parabola’s horizontal and vertical sections. By taking the parabola’s vertices, one can find the parabola’s surface that lies along the chosen section. Another important aspect of choosing a surface for what is a cross-sectional study is the choice of the parabola’s topography. The chosen surface must lie in a region where lighting would be adequate. A good choice would be a flat surface, but there are times when more precision is required. In these cases, special lighting equipment is available.