How do i create points along a spline???

Hello how do i create the Yellow dots(segmentPoints) all around the spline like in Pic1 and draw a blue line to each yellow dot. Instead of just creating the yellow dots in between the white dots(controlPoints) Pic2, would anyone know how to do this?????

-Pic1

-Pic2

Here is my code so you know what i have done.

  public class CatmullSpline : MonoBehaviour {
  
      public List<Vector3> controlPointsList = new List<Transform>();
       List<Vector3> segmentPoints = new List<Vector3>();
   
       public bool isLooping = true;
   
       public int SEGMENT_COUNT;
       void Start()
       {
           CreateSegmentList();
       }
       void OnDrawGizmos()
       {
           for (int i = 0; i < controlPointsList.Count; i++)
           {
   
               if ((i == 0 || i == controlPointsList.Count - 2 || i == controlPointsList.Count - 1) && !isLooping)
               {
                   continue;
               }
   
               DisplayCatmullRomSpline(i);
           }
           Gizmos.color = Color.white;
   
           for (int i = 0; i < controlPointsList.Count; i++)
           {
               Gizmos.DrawWireSphere(controlPointsList[i], 0.3f);
           }
       }
       Vector3 ReturnCatmullRom(float t, Vector3 p0, Vector3 p1, Vector3 p2, Vector3 p3)
       {
           Vector3 a = 0.5f * (2f * p1);
           Vector3 b = 0.5f * (p2 - p0);
           Vector3 c = 0.5f * (2f * p0 - 5f * p1 + 4f * p2 - p3);
           Vector3 d = 0.5f * (-p0 + 3f * p1 - 3f * p2 + p3);
   
           Vector3 pos = a + (b * t) + (c * t * t) + (d * t * t * t);
   
           return pos;
       }
   
       void DisplayCatmullRomSpline(int pos)
       {
   
           Vector3 p0 = controlPointsList[ClampListPos(pos - 1)];
           Vector3 p1 = controlPointsList[pos].position;
           Vector3 p2 = controlPointsList[ClampListPos(pos + 1)];
           Vector3 p3 = controlPointsList[ClampListPos(pos + 2)];
   
           Vector3 lastPos = Vector3.zero;
   
   
           for (float i = 0; i < SEGMENT_COUNT; i++)
           {
               float t = (float)i / SEGMENT_COUNT;
   
               Vector3 newPos = ReturnCatmullRom(t, p0, p1, p2, p3);
   
               if (t == 0)
               {
                   lastPos = newPos;
                   continue;
               }
   
               Gizmos.color = Color.white;
               Gizmos.DrawLine(lastPos, newPos);
   
               Gizmos.color = Color.yellow;
               var tan = ReturnCatmullRomTangent(t, p0, p1, p2, p3);
               Gizmos.DrawLine(newPos, newPos + tan * 3);
               Gizmos.DrawSphere(newPos, 0.3f);
               lastPos = newPos;
           }
   
           Gizmos.DrawLine(lastPos, p2);
       }
       void CreateSegmentList()
       {
           for (int i = 0; i < controlPoints.Count; i++)
           {
               Vector3 p0 = controlPoints[ClampListPos(i - 1)];
               Vector3 p1 = controlPoints[i];
               Vector3 p2 = controlPoints[ClampListPos(i + 1)];
               Vector3 p3 = controlPoints[ClampListPos(i + 2)];
   
   
               for (int j = 1; j < SEGMENT_COUNT; j++)
               {
                   float t = (float)j / (float)SEGMENT_COUNT;
                   segmentPoints.Add(ReturnCatmullRom(t, p0, p1, p2, p3));
   
               }
           }
       }
       int ClampListPos(int pos)
       {
           if (pos < 0)
           {
               pos = controlPointsList.Count - 1;
           }
   
           if (pos > controlPointsList.Count)
           {
               pos = 1;
           }
           else if (pos > controlPointsList.Count - 1)
           {
               pos = 0;
           }
   
           return pos;
       }
   }

@b1gr4yn