Principles of Longitudinal Profiles and Conical Projections

Longitudinal Profile: Analyzing Cutting Contours

Example: Consider a horizontal area in a waterway. It is clear that the profile should show a sharp break rather than a continuous stretch. To correct this, we define a break in the ground using breaklines.

System Representation of Height

To represent height, we utilize geometric tools and random points known as "stitches," each assigned a specific height. The density of these points determines the accuracy of the model.

Accuracy and Human Perception

The human eye can perceive a difference of 0.25 mm with an error margin of 0.2 mm. Consequently, the visible error depends on the map scale. For example, what is the visible error at a scale of 1:50,000?

Distance Definitions

• Dn (Natural Distance): The stretch measured between two points along the profile.
• Dg (Geometrical Distance): The length of the straight line connecting two points.
• Dr (Reduced Distance): The distance projected onto the horizontal plane.

Conical Projection Transformations

Conical projection is performed on the surface of a cone tangent to the sphere. Distortion is minimal in the contact area and increases as one moves away from it.

1. It is ideal for middle latitudes and can use one or two standard parallels.
2. Parallels are arcs of variable spacing, clustered toward the center of the map.
3. Meridians are equally spaced rays that intersect the 90th parallel.
4. It is suitable for countries and regions with significant east-west extent.
5. The latitude range should not exceed 35°.
6. The scale is true along the standard parallel(s).
7. The pole near the standard parallel is a point, while the opposite pole is at infinity.

Azimuthal Projections

1. These involve projecting the globe onto a tangent or secant plane located anywhere on the sphere.
2. Projections must pass through the center of the globe, ensuring distortions are symmetrical around the selected center point.
3. They are used for small surfaces where Earth's sphericity is negligible, such as topographic surveys.
4. The resulting representation is called a plane, commonly used in surveying.