You can find the normal component to the curve at each point. By adding or substracting the normal vector to the position vector of that point, you''ll get another point. Now, do this for each point, and you should get a parallel curve... Even better, from the inital curve, create two parallel curves (so the inital shape serves as an axis).

How to do this? Well a Bézier curve is a R² Bernstein polynomial, of the form

{ X(t) = an t^n + .... a1 t + a0
{ Y(t) = bn t^n + .... b1 t + b0

So the tangent vector is obtained by derivating

{ Xtan(t) = n an t^(n-1) + ..... 2 a2 t + a1
{ Ytan(t) = n bn t^(n-1) + ..... 2 b2 t + b1

So Xnor = -Ytan, Ynor = Xran is a normal vector. Normalize it to have a magnitude vector, and multiply it by half the width of the path you want to create.

Basically, one side of your path will be

{ X(t) + (Width / 2) * (normalized)Xnor(t)
{ Y(t) + (Width / 2) * (normalized)Ynor(t)

And the other will be

{ X(t) - (Width / 2) * (normalized)Xnor(t)
{ Y(t) - (Width / 2) * (normalized)Ynor(t)

So for each (t) you can compute a point on each side of your road.


ToohrVyk

yeah, you need to derive the curve to get the direction at the point of interest on the curve, and the second derivative will give you the ''acceleration'', which would be a direction pointing towards the ''inside'' of the curve. You cross product the two, and it will give you the vector towards the side of the curvature, basically, where to draw the parallel curve. The catmul-romms are not continuous in the second derivative, so they won;t be very useful for what you need. However, since you are doing roads, you can expect the road to be generally pointing up towards the sky. In that case, you can use just the first derivative, and deduce the direction of the width of the road.

// normal spline calculation, to get the point on a spline and the direction at that point.void CalculatePointOnPath(float u, Vector& P, Vector& Dir){...............}Vector CalculateSideVector(Vector Dir, Vector Up=Vector(0, 1, 0)){     Vector Side = Up.Cross(Dir);         Side.Normalise();     Up = Dir.Cross(Side);     return Side;}

so for a given point on the spline, the point on the other lane would be

float u;        // Position on the spline of the control pointVector P;       // point on the reference spline (control point)Vector Dir;     // the direction of the spline at point PVector Up;      // the reference direction of the roadVector Side;    // where the parallel spline should be relative to the reference spline// calculate the point on path and the direction of the path at that pointCalculatePointOnPath(u, P, Dir);Dir.Normalise();Up = Vector(0, 1, 0);// calculate the side direction at the point on path to generate a point on a parallel splineSide = CalculateSideVector(Dir, Up);// calculate the point on the parallel splinePother = P + Side * widthOfLane;

if the spline is C2 continous, you can do

float u;        // Position on the spline of the control pointVector P;       // point on the reference spline (control point)Vector Dir;     // the direction of the spline at point PVector Up;      // the reference direction of the roadVector Side;    // where the parallel spline should be relative to the reference spline// calculate the point on path and the direction of the path at that pointCalculatePointOnPath(u, P, Dir, Up);Dir.Normalise();Up.Normalise();// calculate the side direction at the point on path to generate a point on a parallel splineSide = Up.Cross(Up, Dir);Side.Normalise();// calculate the point on the parallel splinePother = P + Side * widthOfLane;

Id'' advise you to use several 4-point curves (even 3-point ones) because that solution is less computationnally expensive/

ToohrVyk