So, what are the factors that determine how road builders negotiate such natural obstacles that get in the way?
1. Cost
2. Convenience
3. Laws and Regulations
4. Efficiency of Finished Road
5. Safety Considerations
6. Travel Time Saved by Choosing One Option over Another
7. Other Factors
A retired US Engineer's thoughts:
The location and design of roads and rails is an iterative process done in phases. The planning phase involves developing several alternative alignments including a do nothing alternative. The environmental and social impacts, construction costs and constructability of each are identified and compared. Economic analysis is also performed to help determine the most cost effective solution, e.g., go around it, through it or over it. The best components of each alignment alternative that minimize impacts, minimize cost and have desirable constructability are merge into a final alignment alternative. The final alignment is taken into the design phase which includes both preliminary design and final design. Final design plans are used for construction. Public roads are planned and designed to meet demand 20 years in the future. The demand volume 20 years in the future determines the number of lanes. New rail location and design projects are rare. Existing track alignments and right-of-way are almost always used so the planning phase is skipped. Private railroads are profit oriented and therefore planned improvements are for a much shorter timeframe. I am told railroad business plans rarely go beyond 5 years. An economic cost-benefit analysis is performed that includes construction costs, operating costs, vs revenue benefits and time savings benefits. When the benefits are greater than the costs a single track gets more sidings or a single track with sidings gets double tracked. Safety is the primary reason for converting at-grade crossings to grade separated crossings.
1. Cost: Cost is the primary decision factor for roads and rails. Environmental regulations may require a more costly design to avoid an environmental impact. Costs for construction, right-of-way, earth work, bridges and material costs i.e. concrete vs steel vs timber as well as maintenance costs are all taken into account. In urban areas there is also consideration given to road bridge aesthetics. Some times a 'signature' bridge which has a higher cost is acceptable to make it a unique landmark.
2. Convenience: Not sure what you mean by convenience?
3. Laws and Regulations: Environmental regulations and design standards dictate road and rail design for the desired design speed. Horizontal and vertical curves and grades as mentioned above are different for different types of roads, two-lane two-way local roads vs 2-lane oneway freeways and rail. Design speed dictates minimum horizontal and vertical curves and super elevation. For roads stopping horizontal and vertical sight distance is also a factor. If there is an object on the road a driver needs to be able to see it in time to stop. I am sure you are familiar rail design maximum grades, curves and superelevation. At higher speeds railroads use spiral curves between tangent track and curved track sections to ease the rolling stock into and out of the curve. Roadways do not use spiral curves but do have minimum tangent sections between curves, and reverse curves are a no no. for driver safety.
4. Efficiency of Finished Road or Rail: As you said above the shortest distance between two points is a straight line and building something as straight and as flat as the design standards allow yields the lowest construction costs and also the lowest operating costs. Economic analysis of alternatives is common practice.
5. Safety Considerations: Covered by #3 and #4
6. Travel time: Not directly considered in new highway or new rail designs but is considered for certain types of highway improvements. Travel time or rather delay is considered in the rail decision to double track.
7. Other factors: You covered them all pretty well.
Terrain options:
In rolling terrain the alignment is usually straight or slightly curved to balanced cuts and fills to lower construction costs.
In mountainous terrain a curved alignment follows the elevation of the mountain topography along the valley. If that is impractical to get to the other side of the mountain a tunnel is bored. Bridges are built to span a valley when the alignment goes between mountains.
Natural obstacles:
The decision to bridge a river or tunneling under it, go around the mountain or through it, go over a lake or around it depend a lot on the alignment and the type, size, elevation or grade change, location of the obstacle, and the cost of alternatives.
Gosh I really didn't mean to write a thesis on this. Can you tell I miss my work?