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A Euclidean circle with center $c\in {\mathbb{R}}^2$ and radius $r>0$ is the set of points in the Euclidean plane at Euclidean distance exactly $r$ from $c$:It is the boundary curve, while the closed disk includes all points with $\Vert x-c\Vert \le r$. The center belongs to that closed disk but not to the Euclidean circle.

For example, center $c=(2,-1)$ and radius3 give $(x-2{)}^2+(y+1{)}^2=9$. The point $(5,-1)$ lies on the Euclidean circle; $(2,1)$ lies inside the closed disk because its distance from the center is2. Such an equation distinguishes a distance constraint from a region-filling inequality.

Every Euclidean circle is a circle in the topological sense. The map $u\mapsto c+ru$ from the unit circle $S^1$ is a homeomorphism, with inverse $x\mapsto (x-c)/r$. Topology forgets the particular center and radius, whereas the Euclidean distance retains them. The assumption $r>0$ matters: radius zero gives a point, which is not a circle.

The historical terminology differs slightly. Euclid's definition describes the plane figure contained by a circumference; the modern set above corresponds to that boundary. His equal-distance condition and distinguished center appear in BookI, Definitions15–18.^[ref]

The radial sampling interval $\Delta r$ is the separation between consecutive radii at which a radial measurement is made. For Sholl analysis, it determines the spacing of concentric Euclidean circles used to sample intersection counts about a fixed center. The documentation distinguishes this step from the starting and ending radii.^[ref]

For constant positive spacing, the sampling grid iswhere $R$ is the chosen upper radius. For $r_{\mathrm{start}}=1$, $\Delta r=1$ and $R=4.4$, the sampled radii are 1, 2, 3 and 4. The upper bound 4.4 is not itself sampled. The software documentation explicitly notes this endpoint issue when the step does not divide the difference between ending and starting radii.^[ref]

A purely geometric example shows what coarse spacing can miss. Begin with a straight line segment from $(0,0)$ to $(4,0)$ and use the origin as center. Add a short branch from $(2.1,0)$ to $(2.4,0.7)$. Its distance from the center increases continuously from $2.1$ to $2.5$, since $2.4^2+0.7^2=2.5^2$. Counting shared endpoints only once, the added branch contributes one extra distinct intersection exactly forAt radii 1, 2, 3 and 4, the branched and unbranched traces both give $N(r)=1$. An additional sample at $r=2.25$ instead gives two intersections for the branched trace. The feature exists in the geometric input but falls between the coarse sample radii.

Smaller $\Delta r$ supplies a denser set of probes, while changing $r_{\mathrm{start}}$ shifts their positions. Neither operation restores structure already lost through image segmentation or finite pixel pitch. In particular, closer numerical spacing is distinct from improved physical spatial resolution. Report the center, calibrated distance units, starting radius, interval and sampled endpoint list with a Sholl profile.

The program's zero step setting is a special request for finite image-derived sampling, rather than the literal choice $\Delta r=0$ in the grid above. For image input, the documentation describes voxel-scale increments and a calibration-dependent rule for unequal pixel dimensions.^[ref]

Sholl analysis summarizes the radial arrangement of a traced structure by counting its intersections with concentric Euclidean circles around a chosen center. The resulting Sholl profile reports how the count changes with radius. A computational workflow must specify the center, spatial calibration, sampled radii and the convention used to identify distinct intersections. The software documentation distinguishes analysis of segmented images from analysis of reconstructed tracings.^[ref]

For an original center-choice example, take the straight segment $\Gamma =\{(x,0):0\le x\le 2\}$. With center $(0,0)$, a circle of radius $r$ meets the segment once for $0<r\le 2$. With center $(1,0)$, it meets the same segment twice for $0<r\le 1$ and not at all for $r>1$. At $r=0.75$, therefore, the two center choices give counts 1 and 2, although the traced object has not changed. The center is part of the measurement definition.

For binary images, the foreground also needs an explicit meaning. The SNT documentation treats zero-valued pixels as background independently of the displayed lookup table and illustrates how an inverted segmentation can make the program sample background instead of structure.^[ref] Record the foreground-background polarity and radial sampling interval with the result.

The radial summary is useful for comparing where intersections occur, but a Sholl profile does not determine rooted tree topology. A fitted curve, sampled count sequence and complete ideal Sholl profile are different data objects; none should be silently substituted for another.