Conformal Point and the Calibrated Conic

TL;DR

The paper presents two geometric constructs—the calibrating conic and the conformal point—that overlay images to visualize camera calibration and image geometry, providing a real, intuitive alternative to the conventional calibration matrix and the imaginary IAC. It develops purely geometric methods to measure angles and field of view, including a cross-ratio based $\cos^2(\theta)$ formula and a conformal-point construction for angle preservation, all derived from the calibrating conic via $\mathbf{C} = \mathbf{K}^{-\top} \mathbf{D} \mathbf{K}^{-1}$ with $\mathbf{D}=\mathrm{diag}(1,1,-1)$. The framework is extended to practical tasks: calibrating images of tiled floors, outdoor scenes, and paintings, and to odometry under planar motion, where horizon and conformal point remain fixed and pairwise rotations are estimated efficiently (e.g., via two-point RANSAC). By marrying elementary Euclidean geometry with projective concepts, the work offers a visually accessible, algebra-free pathway to camera calibration, angle measurement, and motion estimation relevant to vision systems and graphical applications in the deep-learning era.

Abstract

This gives some information about the conformal point and the calibrating conic, and their relationship one to the other. These concepts are useful for visualizing image geometry, and lead to intuitive ways to compute geometry, such as angles and directions in an image.

Figures (27)

• Figure 1: For images with square pixels, the calibrating conic is a circle with centre at the principal point of the image. If the pixels are rectangular with non-square aspect ration, then the calibrating conic is an axis-aligned ellipse. Finally if skew is present, the conic is an non-axis-aligned ellipse from which one can read the skew parameter as shown.
• Figure 2: Construction of the polar of a point $\hbox{\bf $\bf x$}$ with respect to a conic. The red line is the polar of point $\hbox{\bf $\bf x$}$, as described in the text.
• Figure 3: (Diagram taken from HartleyZisserman:03 figure 8.28). Finding the directions perpendicular to a given direction. Given point $\hbox{\bf $\bf x$}$, the line $\hbox{\bf $\bf l$}$ is the reflected polar of $\hbox{\bf $\bf x$}$ with respect to the calbrating conic $\hbox{\bf$\tt C$}$. Any point on the line $\hbox{\bf $\bf l$}$ represents a direction perpendicular to $\hbox{\bf $\bf x$}$.
• Figure 4: The construction to compute the angle $\theta$ between two points $\hbox{\bf $\bf x$}_1$ and $\hbox{\bf $\bf x$}_2$, using the calibrating conic. Lines $\hbox{\bf $\bf l$}_1$ and $\hbox{\bf $\bf l$}_2$ are the reflected polars of $\hbox{\bf $\bf x$}_1$ and $\hbox{\bf $\bf x$}_2$. They intersect the line joining $\hbox{\bf $\bf x$}_1$ and $\hbox{\bf $\bf x$}_2$ at points $\hbox{\bf $\bf x$}_1'$ and $\hbox{\bf $\bf x$}_2'$ respectively. Then $\cos^2(\theta)$ is given by the cross ratio (\ref{['eq:angle-formula']}). In addition, $\cos(\theta)$ is negative, because $\hbox{\bf $\bf x$}_1$ and $\hbox{\bf $\bf x$}_2$ lie on opposite sides of $\hbox{\bf $\bf l$}_1$.
• Figure 5: Finding the calibrating conic given three orthogonal vanishing points. This diagram is taken directly from HartleyZisserman:03. It shows how the calibrating conic is at the orthocentre of the triangle formed by three vanishing points, $\hbox{\bf $\bf v$}_1$, $\hbox{\bf $\bf v$}_2$ and $\hbox{\bf $\bf v$}_3$. The radius is defined by the condition that the reflected polar of (say) $\hbox{\bf $\bf v$}_1$ passes through $\hbox{\bf $\bf v$}_2$ and $\hbox{\bf $\bf v$}_3$.

Theorems & Definitions (3)

• theorem 2.1
• theorem 2.2
• proof