Recall a Catmull-Rom spline is a combination of Hermite cubic curves joined together at the endpoints. This slide describes how we can smoothly join together curves (three in this example) and answers the question “how do we ensure that the end of the first curve has the same tangent as the beginning of the second curve?”. A solution that was discovered by Catmull and Rom is to simply subtract the previous point from the next point and divide by 2. They discovered that this ensures the curves looks aesthetically pleasing. Thus, the only the thing you need to calculate the tangents at any point (that’s not the start or end) is to obtain the adjacent points.
Recall a Catmull-Rom spline is a combination of Hermite cubic curves joined together at the endpoints. This slide describes how we can smoothly join together curves (three in this example) and answers the question “how do we ensure that the end of the first curve has the same tangent as the beginning of the second curve?”. A solution that was discovered by Catmull and Rom is to simply subtract the previous point from the next point and divide by 2. They discovered that this ensures the curves looks aesthetically pleasing. Thus, the only the thing you need to calculate the tangents at any point (that’s not the start or end) is to obtain the adjacent points.