The Role of the Deltoidal Surface in the Solution Variation of the P3P Problem

Abstract

Traditionally the danger cylinder is intimately related to the solution stability in P3P problem. In this work, we show that the danger cylinder is also closely related to the multiple-solution phenomenon. More specifically, we show that when the optical center lies on the danger cylinder, of the 3 possible P3P solutions, i.e., one double solution, and two other solutions, the optical center of the double solution still lies on the danger cylinder, but the optical centers of the other two solutions no longer lie on the danger cylinder. And when the optical center moves on the danger cylinder, accordingly the optical centers of the two other solutions of the corresponding P3P problem form a new surface, characterized by a polynomial equation of degree 12 in the optical center coordinates, called the deltoidal surface of danger cylinder (DSDC). This indicates the danger cylinder always has a companion deltoidal surface. For the significance of DSDC, we show that when the optical center passes through the DSDC, the number of solutions of P3P constraint system must change by 2, or DSDC acts as a delimitating surface of the P3P solution space. These new findings shed some new lights on the P3P multi-solution phenomenon, an important issue in P3P study.

Appendix (About the Proof of Proposition 2)

The 3 steps of the proof in Sect. 3.2 are detailed below:

Step 1

Around $\left( {{s_{1d}},{s_{2d}},{s_{3d}}} \right)$, the first order differential relationship between $\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\phi _d}}\\ {\mathrm{d}{\varphi _d}}\\ {\mathrm{d}{\eta _d}} \end{array}} \right]$ and $\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{s_{1d}}}\\ {\mathrm{d}{s_{2d}}}\\ {\mathrm{d}{s_{3d}}} \end{array}} \right]$ is:

(A1)

As stated by Rieck [18], when the optical center lies on the danger cylinder, the rank of $J_{11}$ is 2. So, by SVD decomposition of $J_{11}$:

$$\begin{aligned} {J_{11}} = \left[ {\begin{array}{*{20}{c}} {{u_{11}}}&\quad {{u_{21}}}&\quad {{u_{31}}} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} {{\lambda _{11}}}&{}\quad &{}\quad \\ &{}\quad {{\lambda _{21}}}&{}\quad \\ &{}\quad &{}\quad 0 \end{array}} \right] \left[ {\begin{array}{*{20}{c}} {v_{21}^T}\\ {v_{21}^T}\\ {v_{31}^T} \end{array}} \right] \end{aligned}$$

(A2)

Substituting (A2) to (A1), we have:

$$\begin{aligned} \left[ \! {\begin{array}{*{20}{c}} {u_{21}^T}\\ {u_{21}^T}\\ {u_{31}^T} \end{array}}\! \right] \left[ \! {\begin{array}{*{20}{c}} {\mathrm{d}{\phi _d}}\\ {\mathrm{d}{\varphi _d}}\\ {\mathrm{d}{\eta _d}} \end{array}}\! \right] = \left[ \! {\begin{array}{*{20}{c}} {{\lambda _{11}}}&{}\quad &{}\quad \\ &{}\quad {{\lambda _{21}}}&{}\quad \\ &{}\quad &{}\quad 0 \end{array}}\! \right] \left[ \! {\begin{array}{*{20}{c}} {v_{11}^T}\\ {v_{21}^T}\\ {v_{31}^T} \end{array}}\! \right] \left[ \! {\begin{array}{*{20}{c}} {\mathrm{d}{s_{1d}}}\\ {\mathrm{d}{s_{2d}}}\\ {\mathrm{d}{s_{3d}}} \end{array}}\! \right] \begin{array}{*{20}{c}}&\,&\,&\end{array} \end{aligned}$$

(A3)

From Eq. (A3), we have the following equation:

$$\begin{aligned} u_{31}^T\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\phi _d}}\\ {\mathrm{d}{\varphi _d}}\\ {\mathrm{d}{\eta _d}} \end{array}} \right] = 0 \end{aligned}$$

Then we define new terms $f_1,f_2,f_3$, by:

$$\begin{aligned} \left[ {\begin{array}{*{20}{c}} {{f_1}}\\ {{f_2}}\\ {{f_3}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {u_{21}^T}\\ {u_{21}^T}\\ {u_{31}^T} \end{array}} \right] \! \!\left[ {\begin{array}{*{20}{c}} \phi \\ \varphi \\ \eta \end{array}} \right] \end{aligned}$$

By defining: $\left[ {\begin{array}{*{20}{c}} {{\rho _1}}\\ {{\rho _2}}\\ {{\rho _3}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {v_{11}^T}\\ {v_{21}^T}\\ {v_{31}^T} \end{array}} \right] \! \!\left[ {\begin{array}{*{20}{c}} {{s_1}}\\ {{s_2}}\\ {{s_3}} \end{array}} \right]$ we have:

$$\begin{aligned} \left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _1}}\\ {\mathrm{d}{\rho _2}}\\ {\mathrm{d}{\rho _3}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {v_{11}^T}\\ {v_{21}^T}\\ {v_{31}^T} \end{array}} \right] \! \! \left[ {\begin{array}{*{20}{c}} {\mathrm{d}{s_1}}\\ {\mathrm{d}{s_2}}\\ {\mathrm{d}{s_3}} \end{array}} \right] \end{aligned}$$

For ${f_i}({i = 1,2,3})$, around $\left( {{s_{1d}},{s_{2d}},{s_{3d}}} \right)$ by making a second order differential approximation, we have:

$$\begin{aligned} \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {\mathrm{d}{f_{1d}}}\\ {\mathrm{d}{f_{2d}}}\\ {\mathrm{d}{f_{3d}}} \end{array}} \right] \approx \left[ {\begin{array}{*{20}{c}} {\frac{{\partial {f_1}}}{{\partial {\rho _1}}}}&{}\quad {\frac{{\partial {f_1}}}{{\partial {\rho _2}}}}&{}\quad {\frac{{\partial {f_1}}}{{\partial {\rho _3}}}}\\ {\frac{{\partial {f_2}}}{{\partial {\rho _1}}}}&{}\quad {\frac{{\partial {f_2}}}{{\partial {\rho _2}}}}&{}\quad {\frac{{\partial {f_2}}}{{\partial {\rho _3}}}}\\ {\frac{{\partial {f_3}}}{{\partial {\rho _1}}}}&{}\quad {\frac{{\partial {f_3}}}{{\partial {\rho _2}}}}&{}\quad {\frac{{\partial {f_3}}}{{\partial {\rho _3}}}} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] \\ \qquad + \left[ {\begin{array}{*{20}{c}} {{{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }^T}{H_{1d}}\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }\\ {{{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }^T}{H_{2d}}\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }\\ {{{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }^T}{H_{3d}}\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] } \end{array}} \right] \\ \quad = \left[ {\begin{array}{*{20}{c}} {{\lambda _{11}}\mathrm{d}{\rho _{1d}} + {{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }^T}{H_{1d}}\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }\\ {{\lambda _{21}}\mathrm{d}{\rho _{2d}} + {{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }^T}{H_{2d}}\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }\\ {{{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }^T}{H_{3d}}\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] } \end{array}} \right] \end{array} \end{aligned}$$

(A4)

Based on Eq. (A4), we further define:

$$\begin{aligned} \begin{array}{l} \left[ {\begin{array}{*{5}{c}} {\mathrm{d}{h_{1d}}}\\ {\mathrm{d}{h_{2d}}}\\ {\mathrm{d}{h_{3d}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\mathrm{d}{f_{1d}} + \mathrm{d}{f_{2d}}}\\ {\mathrm{d}{f_{1d}} - \mathrm{d}{f_{2d}}}\\ {\mathrm{d}{f_{3d}}} \end{array}} \right] \\ \quad {\approx \left[ {\begin{array}{*{20}{c}} {{\lambda _{11}}\mathrm{d}{\rho _{1d}} + {\lambda _{21}}\mathrm{d}{\rho _{2d}} + {{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }^T}\left( {{H_{1d}} + {H_{2d}}} \right) \left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }\\ {{\lambda _{11}}\mathrm{d}{\rho _{1d}} - {\lambda _{21}}\mathrm{d}{\rho _{2d}} + {{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }^T}\left( {{H_{1d}} - {H_{2d}}} \right) \left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }\\ {\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }^T{H_{3d}}{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] } \end{array}} \right] }\\ \quad {= \left[ {\begin{array}{*{20}{c}} {{\lambda _{11}}\mathrm{d}{\rho _{1d}} + {\lambda _{21}}\mathrm{d}{\rho _{2d}} + {{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _1}}\\ {\mathrm{d}{\rho _2}}\\ {\mathrm{d}{\rho _3}} \end{array}} \right] }^T}{H_{1d}}'\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _1}}\\ {\mathrm{d}{\rho _2}}\\ {\mathrm{d}{\rho _3}} \end{array}} \right] }\\ {{\lambda _{11}}\mathrm{d}{\rho _{1d}} - {\lambda _{21}}\mathrm{d}{\rho _{2d}} + {{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _1}}\\ {\mathrm{d}{\rho _2}}\\ {\mathrm{d}{\rho _3}} \end{array}} \right] }^T}{H_{2d}}'\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _1}}\\ {\mathrm{d}{\rho _2}}\\ {\mathrm{d}{\rho _3}} \end{array}} \right] }\\ {\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }^T{H_{3d}}{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] } \end{array}} \right] } \end{array} \end{aligned}$$

(A5)

As shown in Fig. 6, ${O_c}$, with $\left( {{s_{1c}},{s_{2c}},{s_{3c}}} \right)$, lying on DSDC, is another solution to the P3P problem, corresponding to the solution ${O_d}$. For the optical center, around ${O_c}$, we have the differential constraints:

$$\begin{aligned} \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {\mathrm{d}{h_{1c}}}\\ {\mathrm{d}{h_{2c}}}\\ {\mathrm{d}{h_{3c}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\mathrm{d}{g_{1c}} + \mathrm{d}{g_{2c}}}\\ {\mathrm{d}{g_{1c}} - \mathrm{d}{g_{2c}}}\\ {\mathrm{d}{g_{3c}}} \end{array}} \right] \\ \quad \approx J'\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1c}}}\\ {\mathrm{d}{\rho _{2c}}}\\ {\mathrm{d}{\rho _{3c}}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1c}}}\\ {\mathrm{d}{\rho _{2c}}}\\ {\mathrm{d}{\rho _{3c}}} \end{array}} \right] }^T}{H_{1c}}'\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1c}}}\\ {\mathrm{d}{\rho _{2c}}}\\ {\mathrm{d}{\rho _{3c}}} \end{array}} \right] }\\ {{{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1c}}}\\ {\mathrm{d}{\rho _{2c}}}\\ {\mathrm{d}{\rho _{3c}}} \end{array}} \right] }^T}{H_{2c}}'\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1c}}}\\ {\mathrm{d}{\rho _{2c}}}\\ {\mathrm{d}{\rho _{3c}}} \end{array}} \right] }\\ {{{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1c}}}\\ {\mathrm{d}{\rho _{2c}}}\\ {\mathrm{d}{\rho _{3c}}} \end{array}} \right] }^T}{H_{3c}}'\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1c}}}\\ {\mathrm{d}{\rho _{2c}}}\\ {\mathrm{d}{\rho _{3c}}} \end{array}} \right] } \end{array}} \right] \end{array} \end{aligned}$$

(A6)

The difference between Eqs. (A5) and (A6), is that $J'$ is a full rank matrix. So, for any full rank linear transformation of $\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{h_{1c}}}\\ {\mathrm{d}{h_{2c}}}\\ {\mathrm{d}{h_{3c}}} \end{array}} \right]$, the generated 3 new differential constraints must all contain the first order of $\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1c}}}\\ {\mathrm{d}{\rho _{2c}}}\\ {\mathrm{d}{\rho _{3c}}} \end{array}} \right]$. So, the Eq. (A6) can ignore the second order of $\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1c}}}\\ {\mathrm{d}{\rho _{2c}}}\\ {\mathrm{d}{\rho _{3c}}} \end{array}} \right]$, and we have:

$$\begin{aligned} \left[ {\begin{array}{*{20}{c}} {\mathrm{d}{h_{1c}}}\\ {\mathrm{d}{h_{2c}}}\\ {\mathrm{d}{h_{3c}}} \end{array}} \right] \approx J'\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1c}}}\\ {\mathrm{d}{\rho _{2c}}}\\ {\mathrm{d}{\rho _{3c}}} \end{array}} \right] \end{aligned}$$

(A7)

So, we get the conclusion: at the region nearby $\left( {{s_{1c}},{s_{2c}},{s_{3c}}} \right)$, vector $\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{h_{1c}}}\\ {\mathrm{d}{h_{2c}}}\\ {\mathrm{d}{h_{3c}}} \end{array}} \right]$ is linear with vector $\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1c}}}\\ {\mathrm{d}{\rho _{2c}}}\\ {\mathrm{d}{\rho _{3c}}} \end{array}} \right]$.

Step 2

So, when $\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1c}}}\\ {\mathrm{d}{\rho _{2c}}}\\ {\mathrm{d}{\rho _{3c}}} \end{array}} \right]$ approaches ${\left( {0,0,0} \right) ^T}$ in a fixed direction, $\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{h_{1c}}}\\ {\mathrm{d}{h_{2c}}}\\ {\mathrm{d}{h_{3c}}} \end{array}} \right]$ also approaches ${\left( {0,0,0} \right) ^T}$ in a fixed direction, defined as $D = \left[ {\begin{array}{*{20}{c}} {{D_1}}\\ {{D_2}}\\ {{D_3}} \end{array}} \right]$. So, we have:

$$\begin{aligned} \left[ {\begin{array}{*{20}{c}} {\mathrm{d}{h_{1c}}}\\ {\mathrm{d}{h_{2c}}}\\ {\mathrm{d}{h_{3c}}} \end{array}} \right] \approx k\left[ {\begin{array}{*{20}{c}} {{D_1}}\\ {{D_2}}\\ {{D_3}} \end{array}} \right] ,k \rightarrow 0 \end{aligned}$$

(A8)

As Fact (II), when an optical center is nearby $\left( {{s_{1c}},{s_{2c}},{s_{3c}}} \right)$, another triplet satisfying Eq. (1) whose optical center is nearby $\left( {{s_{1d}},{s_{2d}},{s_{3d}}} \right)$, in the complex space, the imaginary part of which can be either nonzero or zero. The following equation always holds:

$$\begin{aligned}\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{h_{1d}}}\\ {\mathrm{d}{h_{2d}}}\\ {\mathrm{d}{h_{3d}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\mathrm{d}{h_{1c}}}\\ {\mathrm{d}{h_{2c}}}\\ {\mathrm{d}{h_{3c}}} \end{array}} \right] \approx k\left[ {\begin{array}{*{20}{c}} {{D_1}}\\ {{D_2}}\\ {{D_3}} \end{array}} \right] ,k \rightarrow 0 \end{aligned}$$

Step 3

So, we have:

$$\begin{aligned}&\left\{ \begin{array}{l} {D_3}\left( {{\lambda _{11}}\mathrm{d}{\rho _{1d}} + {\lambda _{21}}\mathrm{d}{\rho _{2d}} + {{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }^T}{H_{1d}}'\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] } \right) \\ - {D_1}{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] ^T}{H_{3d}}'\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] \approx 0\\ {D_3}\left( {{\lambda _{11}}\mathrm{d}{\rho _{1d}} - {\lambda _{21}}\mathrm{d}{\rho _{2d}} + {{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] }^T}{H_{2d}}'\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] } \right) \\ - {D_2}{\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] ^T}{H_{3d}}'\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] \approx 0 \end{array} \right. \nonumber \\&{D_3}{\lambda _{11}}\mathrm{d}{\rho _{1d}} + {D_3}{\lambda _{21}}\mathrm{d}{\rho _{2d}} + {\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] ^T}{H_{4d}}'\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] \approx 0 \end{aligned}$$

(A9)

$$\begin{aligned}&{D_3}{\lambda _{11}}\mathrm{d}{\rho _{1d}} - {D_3}{\lambda _{21}}\mathrm{d}{\rho _{2d}} + {\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] ^T}{H_{5d}}'\left[ {\begin{array}{*{20}{c}} {\mathrm{d}{\rho _{1d}}}\\ {\mathrm{d}{\rho _{2d}}}\\ {\mathrm{d}{\rho _{3d}}} \end{array}} \right] \approx 0 \end{aligned}$$

(A10)

Since both Eqs. (A9) and (A10) contain $\mathrm{d}{\rho _{1d}}$ and $\mathrm{d}{\rho _{2d}}$, all the higher order terms of $\mathrm{d}{\rho _{1d}}$ and $\mathrm{d}{\rho _{2d}}$ can be ignored in these two equations. And Eqs. (A9) and (A10) can be simplified as:

$$\begin{aligned}&{D_3}{\lambda _{11}}\mathrm{d}{\rho _{1d}} + {D_3}{\lambda _{21}}\mathrm{d}{\rho _{2d}} + {H_{4d}}'\left( {3,3} \right) \mathrm{d}{\rho _{3d}}^2 \approx 0 \end{aligned}$$

(A11)

$$\begin{aligned}&{D_3}{\lambda _{11}}\mathrm{d}{\rho _{1d}} - {D_3}{\lambda _{21}}\mathrm{d}{\rho _{2d}} + {H_{5d}}'\left( {3,3} \right) \mathrm{d}{\rho _{3d}}^2 \approx 0 \end{aligned}$$

(A12)

So, we have:

$$\begin{aligned}&\left\{ \begin{array}{l} {D_3}{\lambda _{11}}\mathrm{d}{\rho _{1d}} + \frac{{\left( {{H_{4d}}'\left( {3,3} \right) + {H_{5d}}'\left( {3,3} \right) } \right) }}{2}\mathrm{d}{\rho _{3d}}^2 \approx 0\\ {D_3}{\lambda _{21}}\mathrm{d}{\rho _{1d}} + \frac{{\left( {{H_{4d}}'\left( {3,3} \right) - {H_{5d}}'\left( {3,3} \right) } \right) }}{2}\mathrm{d}{\rho _{3d}}^2 \approx 0 \end{array} \right. \nonumber \\&\frac{{\mathrm{d}{\rho _{1d}}}}{{\mathrm{d}{\rho _{3d}}^2}} \approx \frac{{\left( {{H_{4d}}'\left( {3,3} \right) + {H_{5d}}'\left( {3,3} \right) } \right) }}{{ - 2{D_3}{\lambda _{11}}}} \approx 0 \end{aligned}$$

(A13)

$$\begin{aligned}&\frac{{\mathrm{d}{\rho _{2d}}}}{{\mathrm{d}{\rho _{3d}}^2}} \approx \frac{{\left( {{H_{4d}}'\left( {3,3} \right) - {H_{5d}}'\left( {3,3} \right) } \right) }}{{ - 2{D_3}{\lambda _{21}}}} \approx 0 \end{aligned}$$

(A14)

Equations (A13) and (A14) indicate $\mathrm{d}{\rho _{1d}}$, $\mathrm{d}{\rho _{2d}}$ and $\mathrm{d}{\rho _{3d}}^2$ are the infinitesimal quantities of the same order.

$$\begin{aligned} \left[ {\begin{array}{*{20}{c}} {\mathrm{d}{h_{1d}}}\\ {\mathrm{d}{h_{2d}}}\\ {\mathrm{d}{h_{3d}}} \end{array}} \right] \approx \left[ {\begin{array}{*{20}{c}} {{\lambda _{11}}\mathrm{d}{\rho _{1d}} + {\lambda _{21}}\mathrm{d}{\rho _{2d}} + {H_{1d}}'\left( {3,3} \right) \mathrm{d}{\rho _{3d}}^2}\\ {{\lambda _{11}}\mathrm{d}{\rho _{1d}} - {\lambda _{21}}\mathrm{d}{\rho _{2d}} + {H_{2d}}'\left( {3,3} \right) \mathrm{d}{\rho _{3d}}^2}\\ {{H_{3d}}'\left( {3,3} \right) \mathrm{d}{\rho _{3d}}^2} \end{array}} \right] \end{aligned}$$

(A15)

From the third equation of (A15), we have the following 2 cases:

Case (I): when $\mathrm{d}{h_{3d}}/{H_{3d}}'\left( {3,3} \right) > 0$, $\mathrm{d}{\rho _{3d}}$ have 2 real solutions;

Case (II): when $\mathrm{d}{h_{3d}}/{H_{3d}}'\left( {3,3} \right) < 0$, $\mathrm{d}{\rho _{3d}}$ have 2 imaginary solutions.

Note that Case (I) gives two real-valued triplets for the P3P problem, while Case (II) gives two complex-valued triplets. This means that when the optical center passes through the DSDC, a pair of real-valued triplets, which satisfy the constraint system (1), become a pair of complex-valued triplets, which also satisfy the constraint system (1), or vice versa, and hence the number of the P3P constraint system solutions always changes by 2. In addition, when the optical center moves on the DSDC, $\mathrm{d}{h_{3d}}/{H_{3d}}'\left( {3,3} \right) = 0$, the 2 triplets degenerate into the same one in this case, which means the optical center lies on the danger cylinder, and a repeated P3P solution occurs.

Hence Proposition 2 is proved.