1:12 Y.Yin et al. basis vectors w1,w2 represent the p-axis and yp-axis,respectively.With p-axis and yp-axis, we further calculate the zp-axis as w3=wi x w2,to establish the principal-plane coordinate system (principal-frame for short).Here,w1,w2,w3 are described in human-frame.While in the principal-frame,we can represent zp-axis,yp-axis,zp-axis as the unit vector(1,0,0)T, (0,1,0)T,(0,0,1)T,respectively.By comparing w1,w2,w3 in human-frame and xp-axis, Up-axis,Zp-axis in principal-frame,we can get the rotation matrix Rhp,which transforms coordinates from human-frame to principal-frame,as shown in Eq.(5)and Eq.(6). 1 00 1 (5) 01 w 7 (6) With the rotation matrix Rhp,we then calculate the projection of each axis of human- frame in principal plane.For convenience,we use ui,iE[1,3]to represent Th-axis,Uh-axis, Zh-axis,respectively.For ui,its coordinates in the principal-frame is qi,where gi=Rhpui. Then,we get the projection v;of qi in the principal plane with M,i.e.,setting the coordinate value in Zp-axis to zero,as shown in Eq.(7). 「100] vi=Mqi= 010 qi (7) 000 As shown in Fig.10(b),Fig.11(b),Fig.12(b),Fig.13(b)and Fig.14(b),we represent the projected axis of h-axis,yh-axis,zh-axis (of human-frame)in the principal plane with black,green,fuchsia dashed line,respectively. 4.2.2 Reference Plane Detection.In Fig.15,we show how to utilize the length of the projected axis to detect the reference coordinate plane.Intuitively,if the projection of axis wi has shortest length in the principal plane,it indicates that the coordinate plane perpendicular to wi has the highest probability of being parallel to the principal plane,and should be selected as the reference coordinate plane for contour calibration.In the following,we will provide the proof for this intuition. As shown in Fig.15,the plane ABCD and the plane EFGH intersect on line CD(i.e., EF).For simplification,we use Op to represent one of the three axes (i.e.,Zh-axis,yh axis,Zh-axis),the plane ABCD represents the corresponding plane perpendicular to O1P (i.e.,yh-Zh plane,Th-zh plane,Th-yh plane),while EFGH represents the principal plane.To obtain the projection of OP,we first extend the line OP to intersect with EFGH at P,then O1P'LCD,because O1P L ABCD.Besides,we make the line O1I⊥CD,then we get OI⊥CD and O1P'⊥CD,thus CD⊥△OIP'.From point O1,we make the line 010,where 01O1 IP'.Then,we obtain that 010LCD and O1OLIP/,thus 010L EFGH.Similarly,from point P,we make the line PO2,where PO2 LIP'and PO2//010.Therefore,002 is the projection of O1P in the principal plane,0102=01P].sin 01. n regard to01and0g,01+02=90°，2+0g=90°，thus01=s,where 03 means the plane included angle between ABCD and EFGH.If 03(or 01)is equal to zero,then the plane ABCD is parallel to the plane EFGH,and ABCD will be selected as reference coordinate plane,in what the projected 2D contour is similar to that in principal plane in high degree. ACM Trans.Sensor Netw.,Vol.1,No.1,Article 1.Publication date:January 2019.1:12 Y. Yin et al. basis vectors 𝜔1, 𝜔2 represent the 𝑥𝑝-axis and 𝑦𝑝-axis, respectively. With 𝑥𝑝-axis and 𝑦𝑝-axis, we further calculate the 𝑧𝑝-axis as 𝜔3 = 𝜔1 × 𝜔2, to establish the principal-plane coordinate system (principal-frame for short). Here, 𝜔1, 𝜔2, 𝜔3 are described in human-frame. While in the principal-frame, we can represent 𝑥𝑝-axis, 𝑦𝑝-axis, 𝑧𝑝-axis as the unit vector (1, 0, 0)𝑇 , (0, 1, 0)𝑇 , (0, 0, 1)𝑇 , respectively. By comparing 𝜔1, 𝜔2, 𝜔3 in human-frame and 𝑥𝑝-axis, 𝑦𝑝-axis, 𝑧𝑝-axis in principal-frame, we can get the rotation matrix 𝑅ℎ𝑝, which transforms coordinates from human-frame to principal-frame, as shown in Eq. (5) and Eq. (6). ⎡ ⎣ 1 0 0 0 1 0 0 0 1 ⎤ ⎦ = 𝑅ℎ𝑝 ⎡ ⎣ 𝜔 𝑇 1 𝜔 𝑇 2 𝜔 𝑇 3 ⎤ ⎦ (5) 𝑅ℎ𝑝 = ⎡ ⎣ 𝜔 𝑇 1 𝜔 𝑇 2 𝜔 𝑇 3 ⎤ ⎦ −1 (6) With the rotation matrix 𝑅ℎ𝑝, we then calculate the projection of each axis of human￾frame in principal plane. For convenience, we use 𝑢𝑖, 𝑖 ∈ [1, 3] to represent 𝑥ℎ-axis, 𝑦ℎ-axis, 𝑧ℎ-axis, respectively. For 𝑢𝑖, its coordinates in the principal-frame is 𝑞𝑖, where 𝑞𝑖 = 𝑅ℎ𝑝𝑢𝑖. Then, we get the projection 𝑣𝑖 of 𝑞𝑖 in the principal plane with 𝑀, i.e., setting the coordinate value in 𝑧𝑝-axis to zero, as shown in Eq. (7). 𝑣𝑖 = 𝑀 𝑞𝑖 = ⎡ ⎣ 1 0 0 0 1 0 0 0 0 ⎤ ⎦ 𝑞𝑖 (7) As shown in Fig. 10(b), Fig. 11(b), Fig. 12(b), Fig. 13(b) and Fig. 14(b), we represent the projected axis of 𝑥ℎ-axis, 𝑦ℎ-axis, 𝑧ℎ-axis (of human-frame) in the principal plane with black, green, fuchsia dashed line, respectively. 4.2.2 Reference Plane Detection. In Fig. 15, we show how to utilize the length of the projected axis to detect the reference coordinate plane. Intuitively, if the projection of axis 𝜔𝑖 has shortest length in the principal plane, it indicates that the coordinate plane perpendicular to 𝜔𝑖 has the highest probability of being parallel to the principal plane, and should be selected as the reference coordinate plane for contour calibration. In the following, we will provide the proof for this intuition. As shown in Fig. 15, the plane 𝐴𝐵𝐶𝐷 and the plane 𝐸𝐹 𝐺𝐻 intersect on line 𝐶𝐷 (i.e., 𝐸𝐹). For simplification, we use −−→𝑂1𝑃 to represent one of the three axes (i.e., 𝑥ℎ-axis, 𝑦ℎ- axis, 𝑧ℎ-axis), the plane 𝐴𝐵𝐶𝐷 represents the corresponding plane perpendicular to −−→𝑂1𝑃 (i.e., 𝑦ℎ − 𝑧ℎ plane, 𝑥ℎ − 𝑧ℎ plane, 𝑥ℎ − 𝑦ℎ plane), while 𝐸𝐹 𝐺𝐻 represents the principal plane. To obtain the projection of 𝑂1𝑃, we first extend the line 𝑂1𝑃 to intersect with 𝐸𝐹 𝐺𝐻 at 𝑃 ′ , then 𝑂1𝑃 ′ ⊥ 𝐶𝐷, because 𝑂1𝑃 ⊥ 𝐴𝐵𝐶𝐷. Besides, we make the line 𝑂1𝐼 ⊥ 𝐶𝐷, then we get 𝑂1𝐼 ⊥ 𝐶𝐷 and 𝑂1𝑃 ′ ⊥ 𝐶𝐷, thus 𝐶𝐷 ⊥ △𝑂1𝐼𝑃′ . From point 𝑂1, we make the line 𝑂1𝑂′ 1 , where 𝑂1𝑂′ 1 ⊥ 𝐼𝑃′ . Then, we obtain that 𝑂1𝑂′ 1 ⊥ 𝐶𝐷 and 𝑂1𝑂′ 1 ⊥ 𝐼𝑃′ , thus 𝑂1𝑂′ 1 ⊥ 𝐸𝐹 𝐺𝐻. Similarly, from point 𝑃, we make the line 𝑃 𝑂2, where 𝑃 𝑂2 ⊥ 𝐼𝑃′ and 𝑃 𝑂2//𝑂1𝑂′ 1 . Therefore, 𝑂′ 1𝑂2 is the projection of 𝑂1𝑃 in the principal plane, |𝑂′ 1𝑂2| = |𝑂1𝑃| · sin 𝜃1. In regard to 𝜃1 and 𝜃3, 𝜃1+𝜃2 = 90∘ , 𝜃2+𝜃3 = 90∘ , thus 𝜃1 = 𝜃3, where 𝜃3 means the plane included angle between 𝐴𝐵𝐶𝐷 and 𝐸𝐹 𝐺𝐻. If 𝜃3 (or 𝜃1) is equal to zero, then the plane 𝐴𝐵𝐶𝐷 is parallel to the plane 𝐸𝐹 𝐺𝐻, and 𝐴𝐵𝐶𝐷 will be selected as reference coordinate plane, in what the projected 2D contour is similar to that in principal plane in high degree. ACM Trans. Sensor Netw., Vol. 1, No. 1, Article 1. Publication date: January 2019