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Fix spelling mistakes (#122)

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David Thomas
2017-10-16 17:04:21 +01:00
committed by Mike Kamermans
parent 7e1cefa73f
commit d395c261bc
19 changed files with 35 additions and 35 deletions
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components/sections/pointvectors3d/content.en-GB.md
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@@ -4,7 +4,7 @@ Before we move on to the next section we need to spend a little bit of time on t
Getting normals in 3D is in principle the same as in 2D: we need to take the normalised tangent vector, and then rotate it by a quarter turn. However, this is where things get that little more complex: we can turn in quite a few directions, so we need to restrict the rotation to the plane that the tangent lies on. That might sound strange: tangents are themselves lines and lines simultaneously lie on an infinite number of planes, so what's up with that?
Well, we know more about the tangent: we also know its rate of change. Think of the Bezier curve as the path of a car. The curve itself tells us the "place in space" at any given time, and the first derivative at any point tells us the "speed of the car at that point". However, we know more: we also know the tangent at "some next moment in time", the second derivative tells us the "accelleration of the car at that point", and if we add the accelleration to the velocity, we know where the car will be "if the curve stopped changing": as long as the curve we're dealing with is not degenerate (that is to say: it isn't actually a pure 2D curve that we simply rotated in 3D) then at any point in time we know two vectors in the same plane, with a third vector in that same plane, and a fourth vector perpendicular that we don't know yet:
Well, we know more about the tangent: we also know its rate of change. Think of the Bézier curve as the path of a car. The curve itself tells us the "place in space" at any given time, and the first derivative at any point tells us the "speed of the car at that point". However, we know more: we also know the tangent at "some next moment in time", the second derivative tells us the "acceleration of the car at that point", and if we add the acceleration to the velocity, we know where the car will be "if the curve stopped changing": as long as the curve we're dealing with is not degenerate (that is to say: it isn't actually a pure 2D curve that we simply rotated in 3D) then at any point in time we know two vectors in the same plane, with a third vector in that same plane, and a fourth vector perpendicular that we don't know yet:
- **t**, the (normalized) vector for the direction of travel at some point B(t),
- **a**, the difference vector between "the tangent here" to what "the tangent at the next point" would be,