Understanding Anamorphosis, UTM Projection, and Topographic Representation
Anamorphosis: Linear, Angle, and Surface
Linear Anamorphosis: The linear anamorphosis factor is the ratio between a linear element dl on a plane and its corresponding element on the Earth dL. When both elements are equal, the linear anamorphosis modulus will drive the value. Lines with a linear modulus equal to unity are called Automeca.
Angle Anamorphosis: The difference between the angle between two directions on the Earth and the same directions on the plane is the magnitude of shear strain. Projections that preserve angles in the transformation are called conformal projections.
Surface Anamorphosis: The ratio of a differential element of area ds on the plane and the corresponding differential element dS on the Earth’s surface is the superficial modulus. Projections that retain surface areas are called equivalent.
Universal Transverse Mercator (UTM) Projection
The Universal Transverse Mercator (UTM) projection uses the International Hayford ellipsoid and the Potsdam fundamental astronomical point (Datum). Geodetic coordinates originate from the Prime Meridian and the Equator. The cartographic representation system is designed as UTM. The projection is a cylindrical development where the axis of the cylinder is contained in the plane of the Equator. The cylindrical surface is tangent to the Earth along a meridian. To decrease linear deformations that increase as we move away from the prime meridian, the ellipsoid is divided into 60 zones, each spanning 6 degrees of longitude. The cylinder is rotated 61 times, so that the cylinder is tangent to the central meridian of each zone. This results in 60 projections, each with the Equator and the central meridian as its reference lines. The projection extends up to 80 degrees North and South. The polar regions use a stereographic projection.
The UTM projection adheres to the following conditions:
- It is compliant (preserves angles).
- It is automec at the tangent meridian (preserves distances).
- The Equator and the central meridian of each zone are straight lines.
- The origin of Cartesian coordinates in the projection is the intersection of the Equator and the central meridian.
Representation of the Land Surface
Map: A map is any representation of the Earth’s surface. Due to its large size and curvature, proper mapping systems are required.
Plan: A plan is the representation of a small surface area that can be considered flat and not curved, without introducing considerable error.
Scale: The scale is the ratio of resemblance between a measurement on the plan and the corresponding measurement on the ground.
The Extent of Visual Perception
The thinnest line our eyes can see is approximately 0.2 mm. This value is called the graphical error. Depending on the scale used, certain features on the ground may not be represented on the plan. For example, at a scale of 1/5000:
0.2 mm x 5000 = 1000 mm = 1 m
Representations Used in Topography
- Dimensioned Drawings: Each point on the surface can be represented by its projection on the plane and its height compared to a reference point or dimension.
- Contours or Contour Lines: A contour line connects points of equal elevation. This is the same as representing the surface by short horizontal planes, projecting the resulting sections onto the plane, and adding a new dimension.
Contour lines must comply with the following general laws:
- Contour lines cannot intersect or merge (except on cliffs, peaks, cornices, or saddle points).
- Contour lines increase or decrease successively and evenly.
- The land between two contour lines is considered to have a uniform slope.
Offset: The offset is the constant distance between the planes corresponding to the contour lines. The land between two contours is considered constant, i.e., it replaces the real relief with a ruled surface (formed by lines) that relies on the two curves.
Interpretation of Plans with Contour Lines: Fundamental Concepts
- A uniform separation between contour lines indicates that the ground has a constant slope.
- When contour lines approach each other at higher elevations and are separated at lower levels, the slope is concave.
- When contour lines approach each other at lower elevations and are separated at higher elevations, the slope is convex.