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If you translate (move without rotation) a copy of the line towards the curve, the first point where the line touches the curve (the tangent to the curve with the slope of the original line) will be the point on the curve closest to the line.

Draw a connecting line from this tangent point to the original line, intersecting that original line at right angles.

Measure the connecting segment. It is the shortest distance.

Vector analysis will give a mathematically strict solution, I do not have the ability to explain this in sufficient detail.

Q: What is the shortest distance between a straight line and a curve?

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You get a curve. If you join them along the shortest [Euclidean] distance between them, you get a straight line.

... is called a Great Circle arc.

When you curve the line you are travelling you are no longer going directly from one point to the other. If you want to go from one point to another you would want to go directly to the second point.

Curve is one possible opposite of straight if straight refers to line that does not bend. Other meanings of straight like honest, heterosexual, and direct have different opposites than curve.

A tangent, but it (a) has nothing to do with croosing (whatever that may mean); and (b) while it does not cross the curve in the neighbourhood of the point of contact - there is no restriction on it crossing the curve at a distance.

Related questions

You get a curve. If you join them along the shortest [Euclidean] distance between them, you get a straight line.

The shortest distance between two points can be a curve when the path follows the geometry of the space, such as on a sphere or a curved surface where a straight line is not possible. This is because the curve minimizes the distance traveled while staying within the constraints of the space's geometry.

Actual distance traveled refers to the total distance covered between two points, taking into account any curves, turns, or detours in the route. Straight line distance, on the other hand, is the shortest distance between two points, ignoring any obstacles or changes in direction that may affect the path taken.

... is called a Great Circle arc.

The shortest distance between two places on the globe is a great circle route. This is the path that follows the curve of the Earth's surface and represents the shortest distance between two points. Any straight line drawn on a globe represents a segment of a great circle route.

When you curve the line you are travelling you are no longer going directly from one point to the other. If you want to go from one point to another you would want to go directly to the second point.

The shortest distance between any 2 points. An ideal zero-width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points. In Euclidean geometry, exactly one line can be found that passes through any two points.A line in math is a straight line that goes forever on each side.

The phrase "The shortest distance between two points is a straight line" is an ancient geometric principle rooted in Euclidean geometry, where a straight line is the shortest path between two points in a flat, two-dimensional space. It is a fundamental concept in mathematics and has been used for centuries in various applications involving distance and efficiency calculations.

The graph of distance vs. time squared will usually be a curve rather than a straight line. This curve represents a non-uniform acceleration or changing velocity over time, as opposed to constant velocity where the graph would be a straight line. The shape of the curve will depend on the specific relationship between distance and time squared in the given scenario.

Straight lines are defined as lines that extend in one direction without curving or bending. In geometry, a straight line is the shortest distance between two points. They have a constant direction and do not change in orientation.

An imaginary line that follows the curve of the Earth is called a great circle. It represents the shortest distance between two points on the Earth's surface and is often used in navigation and mapping.

The distance is straight path from one place to another place. There might be slight difference between the actual distance and the above mentioned distance because of the route chosen.The distance between the above mentioned places is 8,715.02 miles approximately.