Curved slopes can be defined as any combination of mathematical formulas on a graph. This means there are any combination of ways to get feom one point to another - effectively infinite.
For a 2-dimenzional plane, these curves are usually defined as a value (y, vertical axis) that is calculated at each location of another value (x, horizontal axis).
A linear slope or “straight line” is a simple rise over run. For every x units you travel in a direction, your height will change y units. On a 2d plane it is the “most” distance efficient way to get from A to B.
How you define “smoothness” matters… In math land, the linear slope is the smoothest as its curvature never changes. In real life it results in an abrupt stop and start at either end with a constant velocity along the line.
A real life “smoothest path” when changing the Y-value/height of your line involves a cubic or logarithmic slope-in and slope-out of the line, like this bezier curve.. Think of the “steepness” as the speed of your car (how fast your distance changes along the y axis), and the x axis the time you have been driving. Gradually pushing the accelerator on your car until you’re up to speed, coasting in the middle, then gradually apply the brakes until you come to a stop at point 2.
Curved slopes can be defined as any combination of mathematical formulas on a graph. This means there are any combination of ways to get feom one point to another - effectively infinite.
For a 2-dimenzional plane, these curves are usually defined as a value (y, vertical axis) that is calculated at each location of another value (x, horizontal axis).
A linear slope or “straight line” is a simple rise over run. For every x units you travel in a direction, your height will change y units. On a 2d plane it is the “most” distance efficient way to get from A to B.
How you define “smoothness” matters… In math land, the linear slope is the smoothest as its curvature never changes. In real life it results in an abrupt stop and start at either end with a constant velocity along the line.
A real life “smoothest path” when changing the Y-value/height of your line involves a cubic or logarithmic slope-in and slope-out of the line, like this bezier curve.. Think of the “steepness” as the speed of your car (how fast your distance changes along the y axis), and the x axis the time you have been driving. Gradually pushing the accelerator on your car until you’re up to speed, coasting in the middle, then gradually apply the brakes until you come to a stop at point 2.