The appearance of nanoimprint patterns of each column usually exhibits an ellipse shape. For the purpose of illustration, we consider a model constituted by a single circular column. There are six points,

${P}_{i,\text{}i=0~5}$, and their corresponding position vectors,

${r}_{i,\text{}i=0~5}$, and a unit vector,

$\mathit{n}$, to characterize this circular column as shown in

Figure 1, here the situation is similar to the case of square column arrays. Before the oblique operation performance, points

${P}_{i,\text{}i=0~5}$ and their corresponding position vectors

${r}_{i,\text{}i=0~4}$ and

$\mathit{n}$ can be represented as:

Assume we perform the oblique operation on plane

${E}_{\mathrm{model}}$, we can get the values of

$\phi $ and

$\theta $. Thus, the transformation matrix

${U}_{\phi \theta}$ can be established from Equation (10): Then by utilizing this transformation matrix

${U}_{\phi \theta}$, the new positions of points

${P}_{i,\text{}i=0~5}$ are obtained.

and the new position of unit vector

$n$ can also be obtained.

From the points

${p}_{i,\phi \theta ,\text{}i=0~4}$, they can be obtained directly from position vector

${r}_{i,\phi \theta ,\text{}i=0~4}$, and the unit vector

${\mathit{n}}_{\phi \theta}$, we can write down line equations

${L}_{i,\phi \theta ,\text{}i=0~4}\left(r\right)$. The five line equations describe the center line and four characteristic lines of this circular column.

Finally, by solving the equations sets of the five lines,

${L}_{i,\phi \theta ,\text{}i=0~4}\left(r\right)$; and plane

${E}_{\mathrm{pattern}}$;

${S}_{\mathrm{pattern}}\left(r\right)$; as shown in Equations (16) and (20).

The intersection points;

${p}_{i,\text{}\mathrm{int},\text{}i=0~4}\left(r\right)$ are obtained:

where

$i=0~4$, index 0 means the center of nanoimprint pattern (the shape is usually an ellipse) and indices 1~4 denote the ends of long axis and short axis in this ellipse shape (or circle shape) nanoimprint pattern and (

${x}_{5,\phi \theta \text{}},{y}_{5,\phi \theta},{z}_{5,\phi \theta}$) is the new position of tip point of unit vector n after oblique operation performance. To get the whole nanoimprint pattern, two methods can reach this purpose. One is that we choose lots of points constituting completely the perimeter of circle on the top of circular column, as examples, we choose 360 points to describe well the circle on the top of circular column, that is

${P}^{i,\text{}i=0~360+1}$ as shown in

Figure 1. Then, following the same calculation procedures as before, we can obtain the intersection points

${p}^{i,\text{}\mathrm{int},\text{}i=0~360}\left(r\right)$:

where i = 0~360, index 0 means the center of nanoimprint pattern (the shape is usually an ellipse) and indices 1~360 denote the points on the perimeter of the ellipse shape (or circle shape) nanoimprint pattern and (

${x}_{361,\phi \theta},{y}_{361,\phi \theta},{z}_{361,\phi \theta}$) is the new position of tip point of unit vector n after oblique operation performance. Connecting the 360 interaction points that constitute the perimeter of ellipse shape (or circle shape) nanoimprint pattern, the nanoimprint pattern is obtained finally. Another is that when we have obtained the intersection points

${p}_{i,\text{}\mathrm{int},\text{}i=0~4}\left(r\right)$ as shown in Equation (9); where i = 0~4, index 0 means the center of nanoimprint pattern (the shape is usually an ellipse) and indices 1~4 denote the ends of long axis and short axis in this ellipse shape (or circle shape) nanoimprint pattern and (

${x}_{5,\phi \theta},{y}_{5,\phi \theta \text{}},{z}_{5,\phi \theta}$) is the new position of tip point of unit vector n after oblique operation performance. As the points

${p}^{i,\text{}\mathrm{int},\text{}i=1~4}\left(r\right)$ denote the ends of the long axis and short axis in this ellipse shape (or circle shape) nanoimprint pattern, we can get the lengths of long axis, a, and short axis, b, by coordinate pairs (

${p}^{i,\text{}\mathrm{int},\text{}i=1}\left(r\right)$,

${p}^{i,\text{}\mathrm{int},\text{}i=3}\left(r\right)$) and (

${p}^{i,\text{}\mathrm{int},\text{}i=2}\left(r\right)$;

${p}^{i,\text{}\mathrm{int},\text{}i=4}\left(r\right)$):

Finally, the ellipse shape (or circle shape when a = b) nanoimprint pattern can be easily obtained by utilizing intersection points

${p}^{i,\text{}\mathrm{int},\text{}i=0~4}\left(r\right)$ and the lengths of long axis,

a, and short axis,

b, as shown in Equation (21).