Methods
All of the studies I found measured human ability to place a point on a parabola or circle that was defined only by points along its invisible contour, and in the case of the circle its center point. The more comprehensive studies utilized a Wheatstone stereoscope, a dual monitor setup which allowed the subject to perceive images from two screens as if shown on one, and also to align the image; some of the studies were performed on a single screen allowing the subject to place the point.

Locality
One of the more interesting studies I found concluded that human visual interpolation is based almost completely on as few as four of the closest points on the parabola, variance in points further from the missing point had almost no effect on its placement. These findings support the common theory on human visual interpolation – that it is highly local.

Human Interpolation Vocabulary
The interpolation performed in this experiment was found to be fairly similar with a cubic spline, but were most consistent with an algorithm which minimizes the angles between neighboring line segments drawn between points – including the point added by the subject. The researchers also found that the human interpolation curve “library” included quadratic and possibly cubic curves.

Circles
A human can complete a circle with a great degree of precision given a 180 degree or greater arc. This precision is not dependent on having the center of the circle available. However, when less than 180 degrees, 90 for example, are visible, the center of the circle greatly increases precision. The researches hypothesized that this may be a result of two separate mechanisms for curve completion being used, in the case of a larger arc humans simply continue the curvature. This is significantly more difficult with a short arc, and so the mechanism switches to radius calculation, which is greatly aided by the center being visible.

Accuracy
The human interpolation mechanisms act to greatly increase accuracy in prediction of point placement, even when limited biologically. The retinal receptor mosaic at its densest maintains a spacing of 20-30 sec arc between individual cones, but humans are able to place points with accuracy as high as a 2 sec arc through their capacities of interpolation. A sec arc is 1/3600 of a degree, which makes it clear just how accurate our interpolation mechanisms are.

Interpolating sampled contours in 3-D: analyses of variability and bias

Interpolating sampled contours in 3D: perturbation analyses

Role of a circle’s center in visual interpolation

How much of the visual object is used in estimating its position?

Mechanisms of interpolation in human spatial vision