Calculating Right Vector from Two Points

I'm not sure what your problem exactly is. So I describe the entire process in a more or less coarse manner, and we can go into details at the step identified by you.

1.) Sub-divide the spline path into linear segments.

2.) Compute the (perpendicular) sideway vectors at the beginning and end of each segment.

3.) Compute a common vector for each pair of sideway vectors from the end of one segment and the beginning of the next segment. This step is used to avoid a gap on an outside bend or an overlapping on an inside bend, resp.

4.) Compute a sideway displacement of the line segment using a defined (half) width, so that a closed mesh is yielded in.

I'm not sure what your problem exactly is. So I describe the entire process in a more or less coarse manner, and we can go into details at the step identified by you.

1.) Sub-divide the spline path into linear segments.

2.) Compute the (perpendicular) sideway vectors at the beginning and end of each segment.

3.) Compute a common vector for each pair of sideway vectors from the end of one segment and the beginning of the next segment. This step is used to avoid a gap on an outside bend or an overlapping on an inside bend, resp.

4.) Compute a sideway displacement of the line segment using a defined (half) width, so that a closed mesh is yielded in.

What I am trying to do is get away from this:

FVector extendDir = FVector(0, width, 0);

Because it always extends the mesh on the Y axis, I get this result if I move the spline back while moving it left or right:

What I would like to do is have the mesh maintain it's width where ever it is. I don't want it to get squished together, it needs to be the same width at the end as it was before. So I would like to have everything extend according to the right vector's of the points.

1) I have everything sub-divided and I can change how may segments each part is sub-divided into:

2) I've calculated the sideway vectors for each segement (bottom left point of triangle to left top point of triangle). Does this look correct?

// Transform the bottom left point of the triangle and left top point of the triangle by y axis to get the right vector.
FVector v0right = FRotationMatrix(Path->GetWorldRotationAtDistanceAlongSpline(Path->GetDistanceAlongSplineAtSplinePoint(i) + (segmentLength * s))).TransformVector(FVector(0, 1, 0)).SafeNormal().ForwardVector;
FVector v2right = FRotationMatrix(Path->GetWorldRotationAtDistanceAlongSpline(Path->GetDistanceAlongSplineAtSplinePoint(i) + (segmentLength * s) + segmentLength)).TransformVector(FVector(0, 1, 0)).SafeNormal().ForwardVector;

3) How would I compute the common vector?

4) How would I compute a sideway displacement?

Thanks!

Let the spline be divided in segments, so that a sequence of positions

{ p₀, p₁, p₂, ... , p_n }

is given. The difference vector

d_i := p_i - p_i-1, 0 < i <= n

"ties" 2 consecutive points. Its projection onto the ground plane (assuming this is the x-y plane) is

d'_i := ( d_i;x, d_i;y, 0 )

and the belonging sideway vector, i.e. one of its both perpendicular vectors in the plane, normalized, is

s_i := ( d_i;y, -d_i;x, 0 ) / | d'_i |

This could be used to calculate the 4 corner points of a quad with width w for the segment i as

p_i-1, p_i, p_i + s_i * w, p_i-1 + s_i * w

Although each segment gets the same width w, the result looks bad because of the said gaps and overlaps. This is because at any intermediate p_i there are 2 sideway vectors s_i and s_i+1, and they are normally not identical.

Now, a better crossing point would be located somewhere on the halfway vector

h_i := s_i + s_i+1

which is in general no longer perpendicular to one of the both neighboring segments, so that it cannot be simply scaled like

h_i / | h_i | * w

in order to yield in a constant width of the mesh.

Instead, using a scaling like so

v_i := h_i * w / ( 1 + s_i . s_i+1 )

does the trick (if I've done it correctly). It computes to a vector with exemplary lengths

| v_i |_0° = w

| v_i |_90° = 1.414 w

Then the quad for segment 0 has the corners

p₀, p₁, p₁ + v₁, p₀ + s₁ * w

p_n-1, p_n, p_n + s_n * w, p_n-1 + v_n-1

Hey haegarr! Thanks for the help so far, I really appreciate it. Sorry to bug you with all these questions, but I found a way to get the right vector using the points rotation (I didn't know it had a function like that until yesterday) and was wondering if this looks right now:

float distance = Path->GetDistanceAlongSplineAtSplinePoint(i) + (segmentLength * s);
			float distance2 = Path->GetDistanceAlongSplineAtSplinePoint(i) + (segmentLength * s) + segmentLength;

			FVector v0 = GetTransform().InverseTransformPosition(Path->GetWorldLocationAtDistanceAlongSpline(distance));
			FVector v2 = GetTransform().InverseTransformPosition(Path->GetWorldLocationAtDistanceAlongSpline(distance2));

			FRotator v0Rotation = Path->GetWorldRotationAtDistanceAlongSpline(distance);
			FRotator v2Rotation = Path->GetWorldRotationAtDistanceAlongSpline(distance2);

			FVector s1 = FRotationMatrix(v0Rotation).GetScaledAxis(EAxis::Y);
			FVector s2 = FRotationMatrix(v2Rotation).GetScaledAxis(EAxis::Y);

			FVector h = s1 + s2;

			FVector extendDir = h * width / (FVector(1, 1, 1) + s1 * s2);

I was a little confused about calculating v_i though (extendDir is that vector). How I read it was "[halfway vector] multiplied by [width] divided by [a vector with all components set to 1] plus [sideway vector 1] multiplied [sideway vector 2]". I'm pretty sure I didn't read it right. Can you type the equation out in words? lol And are these the right points for each calculation?

Thanks for the help, I really appreciate it again man!

Hey! I got some sleep and I understand it now. Lol I don't know why I was thinking it was a multiplication sign. I changed it to "1 + FVector::DotProduct(s1, s2)" now. :D I will be testing it soon.

Hey haegarr! I got it working thanks to you! Thanks! :D