Very nice! I once implemented the length calculation with the same idea but my code was very ugly and unreadable. Your videos are awesome because your explanations are very clear and straight foreward. Timeless tutorial videos!
There it is, excellent video. I've been neglecting catmull-rom splines for a long time, even though I'm very familiar with bezier curves and parametric surfaces implementantions. I must say you've saved me a lot of effort with these two spline videos.
Huge thanks for these two videos. I always find using graphs, diagrams etc to be crucial for me to understanding abstract things like this and you've certainly helped immensely in doing this - cheers. Really enjoy your videos, gives the old grey matter some needed exercise.
Awesome, I have been looking for something like this for years, I will be using some of your code and concepts in your videos for my 'too big' project. Thanks for doing this, I hope your channel gets the magic youtube boost.
Also, you might want to use binary search to find the node & offset from a point on the spline's perimeter. Of course, this means that the nodes would have to store not only their length, but also their distance from 0.
I implemented a spline class in my game engine and I noticed that it was very difficult to try and get true velocity independence when sampling from the spline using the normalized percentage value t. I think this is due to samples being more concentrated around the end points of each segment and thus the velocity would fluctuate a little between each point. To solve this, I created a function in my spline class that computes a t value from a distance along the spline in world units using a binary search to find the t that is closest to the desired distance along the spline. Then to move along the spline at an independent velocity, I declare a distance variable that is incremented each frame by a velocity value which I then use to sample from the spline data using the previously mentioned function to compute the transform of whatever is traversing the spline.
Yes! It calculates it based on the spline's state when you call the function... though since it uses a binary search function it may be more efficient to pre-compute a linear path from the spline whenever the spline changes rather than every frame.
Found the same problem and then saw your comment. You are right! There is non-linearity in "t" regarding to the distances, so the divide operation in "GetNormalisedOffset" which intended to result the fractional "t" will be inaccurate. Binary search is good idea!
I have to correct you. Although you can use linear rectification to estimate the length of a curve, you can do so while avoiding iteration with elliptic integrals. The lack of sustainable arc length for the ellipsis required the need for the elliptic integral. My research gathered quite a few matlab projects that solve for incomplete arclength in three different orders, or by using Jacobi / Jacobi Zeta in four different ways. Hunt around for moiseevigor's repository and find the computer answer to the arclength for nonclosed incomplete arcs like the hematite curve.
Hey thanks Xavery. I probably will look at DirectX next year. I've used both OpenGL and DX quite a bit, and would like to talk about them, the downside is they require a lot of preparation for the viewer in terms of understanding, so I don't know if they'll fit my format - perhaps a "project" series of videos would be required. So maybe!
i've not looked at my code for ages, but i seem to remember the kink is called an inflection point and they normally appear around t =0.25, t=0.5 and t=0.75.
Quick question: Do you think it would be feasible to differentiate the equations again to get the acceleration? The reason for this is so that if you were modelling an aircraft or spacecraft, say, you could get the model to bank over in the turns.
I really love your 2D implimentations. They helped me learn a lot about interpolation, itself. I think they'd help a lot more if you did a Unity 3D implementation.
Thanks Lupine, You can easily extend this to 3D, 4D and onwards! :D, the maths is applied independent of the axis, so just add a z component in the same style as x and y.
How would apply this to create a spline of rotations -i.e. quaternions? It would allow smoothly going from one rotation to the next, then the next, and so on.
The spline can exist in as many dimensions as you like, as its axis independent, so just interpolate your coefficients. Granted, with quats the output may need some tweaking to ensure continuity but it may just simply work.
@@javidx9 So I scaled then added each quaternion component (w, x, y, z) with the respective coefficient -as you did in the video. I may have misunderstood your written steps above, since there's no smoothness difference from just looping the rotations through a Slerp. This leads me to hope that maybe one day you'd make a video about it -one can hope (U_U)
From file OneLoneCoder_Splines2.cpp if t starts in the loop with 0 then newPoint=oldPoint so the t loop must start with fStepSize float CalculateSegmentLength(int node, bool bLooped = false) { float fLength = 0.0f; float fStepSize = 0.005; sPoint2D old_point, new_point; old_point = GetSplinePoint((float)node, bLooped); for (float t = fStepSize; t < 1.0f; t += fStepSize) { new_point = GetSplinePoint((float)node + t, bLooped); fLength += sqrtf((new_point.x - old_point.x)*(new_point.x - old_point.x) + (new_point.y - old_point.y)*(new_point.y - old_point.y)); old_point = new_point; } return fLength; } DJ
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I really can only join the praise. It's simply amazing how you manage to add so much info while still being that intelligible
Very nice! I once implemented the length calculation with the same idea but my code was very ugly and unreadable. Your videos are awesome because your explanations are very clear and straight foreward. Timeless tutorial videos!
Hey Thanks Carsten!
There it is, excellent video.
I've been neglecting catmull-rom splines for a long time, even though I'm very familiar with bezier curves and parametric surfaces implementantions.
I must say you've saved me a lot of effort with these two spline videos.
Glad I could help Kareshi. Cheers!
Huge thanks for these two videos. I always find using graphs, diagrams etc to be crucial for me to understanding abstract things like this and you've certainly helped immensely in doing this - cheers. Really enjoy your videos, gives the old grey matter some needed exercise.
of all the times i tried diving into splines, this (series of two) made me "get" it. Thanks :D
Nice one Rho, Splines can be a bit difficult, but as soon as you realise just how simple they are, you never stop using them!
You are one of the best teachers i have ever had since i started college
Thanks!
Awesome, I have been looking for something like this for years, I will be using some of your code and concepts in your videos for my 'too big' project.
Thanks for doing this, I hope your channel gets the magic youtube boost.
Thank ProCactus, as long as people find it useful, and you have, then I'm happy!
I love watching that wireframe space ship wind and wend its way around the spline path.
Also, you might want to use binary search to find the node & offset from a point on the spline's perimeter. Of course, this means that the nodes would have to store not only their length, but also their distance from 0.
I implemented a spline class in my game engine and I noticed that it was very difficult to try and get true velocity independence when sampling from the spline using the normalized percentage value t. I think this is due to samples being more concentrated around the end points of each segment and thus the velocity would fluctuate a little between each point. To solve this, I created a function in my spline class that computes a t value from a distance along the spline in world units using a binary search to find the t that is closest to the desired distance along the spline. Then to move along the spline at an independent velocity, I declare a distance variable that is incremented each frame by a velocity value which I then use to sample from the spline data using the previously mentioned function to compute the transform of whatever is traversing the spline.
Hi Ruxify, that's quite an interesting approach. Does it handle the spline changing in real time too?
Yes! It calculates it based on the spline's state when you call the function... though since it uses a binary search function it may be more efficient to pre-compute a linear path from the spline whenever the spline changes rather than every frame.
Found the same problem and then saw your comment. You are right! There is non-linearity in "t" regarding to the distances, so the divide operation in "GetNormalisedOffset" which intended to result the fractional "t" will be inaccurate. Binary search is good idea!
I have to correct you. Although you can use linear rectification to estimate the length of a curve, you can do so while avoiding iteration with elliptic integrals. The lack of sustainable arc length for the ellipsis required the need for the elliptic integral. My research gathered quite a few matlab projects that solve for incomplete arclength in three different orders, or by using Jacobi / Jacobi Zeta in four different ways. Hunt around for moiseevigor's repository and find the computer answer to the arclength for nonclosed incomplete arcs like the hematite curve.
Thanks for the excellent tutorial.
Are there any plans to make video on B-Spline and Bezier curve ?
I really enjoy your series. Thank you for this.
Hi and no problem Mike!
man that was awesome, very simple and clear explanation and everything done in windows console! Do you plan to make any series in OpenGL / DirectX?
Hey thanks Xavery. I probably will look at DirectX next year. I've used both OpenGL and DX quite a bit, and would like to talk about them, the downside is they require a lot of preparation for the viewer in terms of understanding, so I don't know if they'll fit my format - perhaps a "project" series of videos would be required. So maybe!
great to hear that :D
i've not looked at my code for ages, but i seem to remember the kink is called an inflection point and they normally appear around t =0.25, t=0.5 and t=0.75.
Hi WeepingCorpse, I believe you are right.
javidx9 I've been binge watching your channel. I keep wanting to do something like it myself but haven't plucked up the courage to go on camera lol.
Quick question: Do you think it would be feasible to differentiate the equations again to get the acceleration? The reason for this is so that if you were modelling an aircraft or spacecraft, say, you could get the model to bank over in the turns.
Hi Davstr, of course you can keep differentiating! Might be worth a try once I get 3D in the console finished.
Do a tutorial on how to implement NURBS curves.
amazing! thanks for that!
No problem Bruno, thanks!
you might need to make some lookup map for any t for this purpose as splines are calculated once.
I really love your 2D implimentations. They helped me learn a lot about interpolation, itself. I think they'd help a lot more if you did a Unity 3D implementation.
Thanks Lupine, You can easily extend this to 3D, 4D and onwards! :D, the maths is applied independent of the axis, so just add a z component in the same style as x and y.
Very nice! I love how extensible these equations are. Its fascinating.
@@javidx9 Great!! Thiss helpes me a lot!! Thank you so much!!~
Basically just increase the resolution of the spline for speed calculation. I wonder if this is kind of a nyquist thing. Or aliasing, essentially
how do you split a quadratic spline into 2 subsplines please ?
Nice! Now fCurves makes more sense in Blender :D
How would apply this to create a spline of rotations -i.e. quaternions? It would allow smoothly going from one rotation to the next, then the next, and so on.
The spline can exist in as many dimensions as you like, as its axis independent, so just interpolate your coefficients. Granted, with quats the output may need some tweaking to ensure continuity but it may just simply work.
@@javidx9 So I scaled then added each quaternion component (w, x, y, z) with the respective coefficient -as you did in the video. I may have misunderstood your written steps above, since there's no smoothness difference from just looping the rotations through a Slerp. This leads me to hope that maybe one day you'd make a video about it -one can hope (U_U)
From file OneLoneCoder_Splines2.cpp
if t starts in the loop with 0 then newPoint=oldPoint so the t loop must start with fStepSize
float CalculateSegmentLength(int node, bool bLooped = false) {
float fLength = 0.0f;
float fStepSize = 0.005;
sPoint2D old_point, new_point;
old_point = GetSplinePoint((float)node, bLooped);
for (float t = fStepSize; t < 1.0f; t += fStepSize) {
new_point = GetSplinePoint((float)node + t, bLooped);
fLength += sqrtf((new_point.x - old_point.x)*(new_point.x - old_point.x)
+ (new_point.y - old_point.y)*(new_point.y - old_point.y));
old_point = new_point;
}
return fLength;
}
DJ
Thanks DJ, this is a minor, but completely valid optimisation!
Did you mean to say Euclidean distance at 3:34?
No