var measurementData = [
  {
    "id": "Distance_First_Second",
    "name": "Distance between 1st point - 2nd point",
    "imageFile": "measurement_distance_between_two_points-frontal.png",
    "description": "To measure the distance between 2 points that are not listed as feature points in the list: Select '1st Point' then mark the point on the model. Then select '2nd Point' and mark the point on the model. The distance will be calculated.",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(FirstPoint.x-SecondPoint.x\\right)^2+\\left(FirstPoint.y-SecondPoint.y\\right)^2+\\left(FirstPoint.z-SecondPoint.z\\right)^2}$$"
  },
  {
    "id": "Angle_First_Second_Third",
    "name": "Angle between 1st, 2nd & 3rd points",
    "imageFile": "measurement_angle_between_three_points-basal.png",
    "description": "To measure the angle between 3 points that form an angle at the 2nd point: Select '1st Point' then mark the point on the model. Then do the same for '2nd & 3rd Point'. The angle will be calculated.",
    "type": "Angles",
    "formula": "V1=$$<\\left(SecondPoint.x-FirstPoint.x\\right),\\left(SecondPoint.y-FirstPoint.y\\right),\\left(SecondPoint.z-FirstPoint.z\\right)>$$ V2=$$<\\left(SecondPoint.x-ThirdPoint.x\\right),\\left(SecondPoint.y-ThirdPoint.y\\right),\\left(SecondPoint.z-ThirdPoint.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Ratio_First_Second_Third_Fourth",
    "name": "Ratio of two distances [1st to 2nd] / [3rd to 4th]",
    "imageFile": "measurement_ratio_between_two_distances.png",
    "description": "To measure the ratio of two distances. The first distance is between 1st and 2nd points and the second distance is between the 3rd and 4th points.",
    "type": "Ratios",
    "formula": "$$\\frac{\\sqrt{\\left(FirstPoint.x-SecondPoint.x\\right)^2+\\left(FirstPoint.y-SecondPoint.y\\right)^2+\\left(FirstPoint.z-SecondPoint.z\\right)^2}}{\\sqrt{\\left(ThirdPoint.x-FourthPoint.x\\right)^2+\\left(ThirdPoint.y-FourthPoint.y\\right)^2+\\left(ThirdPoint.z-FourthPoint.z\\right)^2}}$$"
  },
  {
    "id": "Ala/Alar_Length_(left)",
    "name": "Ala/Alar Length (left)",
    "imageFile": "measurement_distance_left_alar_length-basal.png",
    "description": "Distance between 'Pronasale/Tip (prn)' and 'Alar Curvature - left (ac_l)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Prn.x-Ac_l.x\\right)^2+\\left(Prn.y-Ac_l.y\\right)^2+\\left(Prn.z-Ac_l.z\\right)^2}$$"
  },
  {
    "id": "Ala/Alar_Length_(right)",
    "name": "Ala/Alar Length (right)",
    "imageFile": "measurement_distance_right_alar_length-basal.png",
    "description": "Distance between 'Pronasale/Tip (prn)' and 'Alar Curvature - right (ac_r)' ",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Prn.x-Ac_r.x\\right)^2+\\left(Prn.y-Ac_r.y\\right)^2+\\left(Prn.z-Ac_r.z\\right)^2}$$"
  },
  {
    "id": "Alar_Base_Width",
    "name": "Alar Base Width",
    "imageFile": "measurement_distance_alar_base_width-basal.png",
    "description": "Distance from 'Alar Curvature/Alar Base Junction - left (ac_l)' to 'Alar Curvature/Alar Base Junction - right (ac_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Ac_l.x-Ac_r.x\\right)^2+\\left(Ac_l.y-Ac_r.y\\right)^2+\\left(Ac_l.z-Ac_r.z\\right)^2}$$"
  },
  {
    "id": "Base_Bony_Width",
    "name": "Base Bony Width",
    "imageFile": "measurement_distance_base_bony_width-frontal.png",
    "description": "Distance between 'Nasal Parenthesis - left (np_l)' and 'Nasal Parenthesis - right (np_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Np_l.x-Np_r.x\\right)^2+\\left(Np_l.y-Np_r.y\\right)^2+\\left(Np_l.z-Np_r.z\\right)^2}$$"
  },
  {
    "id": "Chin_Projection",
    "name": "Chin Projection 1 (Gonzalles-Ulloas)",
    "imageFile": "measurement_distance_chin_projection1-lateral(l).png",
    "description": "The distance from Pogonion (pg) to the Vertical Facial Plane (vfp).",
    "type": "Distances",
    "formula": "MISSING"
  },
  {
    "id": "Columella_Length",
    "name": "Columella Length",
    "imageFile": "measurement_distance_columella_length-lateral(l).png",
    "description": "Distance between 'Subnasale (sn)' and 'Columella Breakpoint (c)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Sn.x-C.x\\right)^2+\\left(Sn.y-C.y\\right)^2+\\left(Sn.z-C.z\\right)^2}$$"
  },
  {
    "id": "Columella_Width",
    "name": "Columella Width",
    "imageFile": "measurement_distance_columella_width-basal.png",
    "description": "Distance from 'Subnasale' left (sn'_l)' to 'Subnasale' right (sn'_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Sn’_l.x-Sn’_r.x\\right)^2+\\left(Sn’_l.y-Sn’_r.y\\right)^2+\\left(Sn’_l.z-Sn’_r.z\\right)^2}$$"
  },
  {
    "id": "Columella-Alar_Distance_-_left",
    "name": "Columella-Alar Distance - left",
    "imageFile": "measurement_distance_columella-alar_distance_left-lateral.png",
    "description": "Distance between 'Alar Ridge's Highest Point - left (armax_l)' and 'Columella's Lowest Point - left (cmin_l)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Armax_l.x-Cmin_l.x\\right)^2+\\left(Armax_l.y-Cmin_l.y\\right)^2+\\left(Armax_l.z-Cmin_l.z\\right)^2}$$"
  },
  {
    "id": "Columella-Alar_Distance_-_right",
    "name": "Columella-Alar Distance - right",
    "imageFile": "measurement_distance_columella-alar_distance_right-lateral.png",
    "description": "Distance between 'Alar Ridge's Highest Point - right (armax_r)' and 'Columella's Lowest Point - right (cmin_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Armax_r.x-Cmin_r.x\\right)^2+\\left(Armax_r.y-Cmin_r.y\\right)^2+\\left(Armax_r.z-Cmin_r.z\\right)^2}$$"
  },
  {
    "id": "Face_Length/Height",
    "name": "Face Length/Height",
    "imageFile": "measurement_distance_facial_height-frontal.png",
    "description": "Distance between 'Trichion (tr)' and 'Menton/Gnathion (me)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Tr.x-Me.x\\right)^2+\\left(Tr.y-Me.y\\right)^2+\\left(Tr.z-Me.z\\right)^2}$$"
  },
  {
    "id": "Face_Width",
    "name": "Face Width",
    "imageFile": "measurement_distance_face_width-frontal.png",
    "description": "Distance from 'Zygion - left (zy_l)' to 'Zygion - right (zy_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Zy_l.x-Zy_r.x\\right)^2+\\left(Zy_l.y-Zy_r.y\\right)^2+\\left(Zy_l.z-Zy_r.z\\right)^2}$$"
  },
  {
    "id": "Height_of_the_Mandible/Chin_Height",
    "name": "Height of the Mandible",
    "imageFile": "measurement_distance_chin_height-lateral(l).png",
    "description": "Distance between 'Menton/Gnathion (me)' and 'Stomion (sto)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Me.x-Sto.x\\right)^2+\\left(Me.y-Sto.y\\right)^2+\\left(Me.z-Sto.z\\right)^2}$$"
  },
  {
    "id": "Infratip_Lobule/Infralobule",
    "name": "Infratip Lobule",
    "imageFile": "measurement_distance_infratip_lobule-lateral(l).png",
    "description": "Distance between 'Pronasale/Tip (prn)' and 'Columella Break Point (c)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Prn.x-C.x\\right)^2+\\left(Prn.y-C.y\\right)^2+\\left(Prn.z-C.z\\right)^2}$$"
  },
  {
    "id": "Interalar_Distance",
    "name": "Interalar Distance",
    "imageFile": "measurement_distance_interalar_distance-frontal.png",
    "description": "Distance between 'Alare/Alar Flare - left (al_l)' and 'Alar/Alar Flare - right (al_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Al_l.x-Al_r.x\\right)^2+\\left(Al_l.y-Al_r.y\\right)^2+\\left(Al_l.z-Al_r.z\\right)^2}$$"
  },
  {
    "id": "(Inner)_Intercanthal_Distance/Width",
    "name": "(Inner) Intercanthal Distance/Width",
    "imageFile": "measurement_distance_intercanthal_distance-frontal.png",
    "description": "Distance between 'Endocanthion/Medial Canthus - left (en_l)' and 'Endocanthion/Medial Canthus - right (en_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(En_l.x-En_r.x\\right)^2+\\left(En_l.y-En_r.y\\right)^2+\\left(En_l.z-En_r.z\\right)^2}$$"
  },
  {
    "id": "Interpupillary/Interocular_Distance",
    "name": "Interpupillary Distance",
    "imageFile": "measurement_distance_interpupillary_distance-frontal.png",
    "description": "Distance from 'Pupil - left (p_l)' to 'Pupil - right (p_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(P_l.x-P_r.x\\right)^2+\\left(P_l.y-P_r.y\\right)^2+\\left(P_l.z-P_r.z\\right)^2}$$"
  },
  {
    "id": "Left_Eye_Width",
    "name": "Left Eye Width",
    "imageFile": "measurement_distance_left_eye_length-frontal.png",
    "description": "Distance from 'Endocanthion/Medial Canthus - left (en_l)' to 'Exocanthion/Lateral Canthus – left (ex_l)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(En_l.x-Ex_l.x\\right)^2+\\left(En_l.y-Ex_l.y\\right)^2+\\left(En_l.z-Ex_l.z\\right)^2}$$"
  },
  {
    "id": "Lower_Facial_Height",
    "name": "Lower Facial Height",
    "imageFile": "measurement_distance_lower_facial_height-frontal.png",
    "description": "Distance between 'Subnasale (sn)' and 'Menton/Gnathion (me)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Me.x-Sn.x\\right)^2+\\left(Me.y-Sn.y\\right)^2+\\left(Me.z-Sn.z\\right)^2}$$"
  },
  {
    "id": "Lower_Lip_Height",
    "name": "Lower Lip Height",
    "imageFile": "measurement_distance_lower_lip_height-frontal.png",
    "description": "Distance from 'Sublabiale (sl)' to 'Stomion (sto)', when the mouth is in a neutral position",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Sl.x-Sto.x\\right)^2+\\left(Sl.y-Sto.y\\right)^2+\\left(Sl.z-Sto.z\\right)^2}$$"
  },
  {
    "id": "Mid-Facial_Height",
    "name": "Mid-Facial Height",
    "imageFile": "measurement_distance_mid_facial_height-frontal.png",
    "description": "Distance between 'Subnasale (sn)' and 'Glabella (g)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(G.x-Sn.x\\right)^2+\\left(G.y-Sn.y\\right)^2+\\left(G.z-Sn.z\\right)^2}$$"
  },
  {
    "id": "Mouth/Lips_Width",
    "name": "Mouth/Lips Width",
    "imageFile": "measurement_distance_mouth_width-basal.png",
    "description": "Distance from 'Cheilion - left (ch_l)' to 'Cheilion - right (ch_r)', when the mouth is in a neutral position.",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Ch_l.x-Ch_r.x\\right)^2+\\left(Ch_l.y-Ch_r.y\\right)^2+\\left(Ch_l.z-Ch_r.z\\right)^2}$$"
  },
  {
    "id": "Nasal_Bridge_Length",
    "name": "Nasal Bridge Length",
    "imageFile": "measurement_distance_nasal_bridge_length-lateral(l).png",
    "description": "Distance between 'Pronasale/Tip (prn)' and 'Nasion (n)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Prn.x-N.x\\right)^2+\\left(Prn.y-N.y\\right)^2+\\left(Prn.z-N.z\\right)^2}$$"
  },
  {
    "id": "Nasal_Root_Width",
    "name": "Nasal Root Width",
    "imageFile": "measurement_distance_nasal_root_width-frontal.png",
    "description": "Distance between 'Maxilloanteriorale - left (mf_l)' and 'Maxilloanteriorale - right (mf_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Mf_l.x-Mf_r.x\\right)^2+\\left(Mf_l.y-Mf_r.y\\right)^2+\\left(Mf_l.z-Mf_r.z\\right)^2}$$"
  },
  {
    "id": "Nasal_Tip_Projection",
    "name": "Nasal Tip Projection",
    "imageFile": "measurement_distance_nasal_tip_projection_I-lateral(l).png",
    "description": "Distance between 'Subnasale (sn)' and 'Pronasale/Tip (prn)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Prn.x-Sn.x\\right)^2+\\left(Prn.y-Sn.y\\right)^2+\\left(Prn.z-Sn.z\\right)^2}$$"
  },
  {
    "id": "Nose_Height",
    "name": "Nose Height",
    "imageFile": "measurement_distance_nose_height-lateral(l).png",
    "description": "Distance between 'Subnasale (sn)' and 'Nasion (n)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Sn.x-N.x\\right)^2+\\left(Sn.y-N.y\\right)^2+\\left(Sn.z-N.z\\right)^2}$$"
  },
  {
    "id": "Nostril_Floor_Width_(left)",
    "name": "Nostril Floor Width (left)",
    "imageFile": "measurement_distance_left_nostril_floor_width-basal.png",
    "description": "Distance between 'Subnasale (sn)' to 'Subalare - left (sbal_l)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Sn.x-Sbal_l.x\\right)^2+\\left(Sn.y-Sbal_l.y\\right)^2+\\left(Sn.z-Sbal_l.z\\right)^2}$$"
  },
  {
    "id": "Nostril_Floor_Width_(right)",
    "name": "Nostril Floor Width (right)",
    "imageFile": "measurement_distance_right_nostril_floor_width-basal.png",
    "description": "Distance between 'Subnasale (sn)' to 'Subalare - right (sbal_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Sn.x-Sbal_r.x\\right)^2+\\left(Sn.y-Sbal_r.y\\right)^2+\\left(Sn.z-Sbal_r.z\\right)^2}$$"
  },
  {
    "id": "Outer_Intercanthal_Distance",
    "name": "Outer Intercanthal Distance",
    "imageFile": "measurement_distance_outercanthal_distance-frontal.png",
    "description": "Distance between 'Exocanthion/Lateral Canthus – left (ex_l)' and 'Exocanthion/Lateral Canthus – right (ex_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Ex_l.x-Ex_r.x\\right)^2+\\left(Ex_l.y-Ex_r.y\\right)^2+\\left(Ex_l.z-Ex_r.z\\right)^2}$$"
  },
  {
    "id": "Philtrum_Width/Interphiltral_Distance",
    "name": "Philtrum Width/Interphiltral Distance",
    "imageFile": "measurement_distance_philtrum_width-basal.png",
    "description": "Distance from 'Crista Philtri - left (cph_l)' to 'Crista Philtri - right (cph_r)' , when the mouth is in a neutral position",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Cph_l.x-Cph_r.x\\right)^2+\\left(Cph_l.y-Cph_r.y\\right)^2+\\left(Cph_l.z-Cph_r.z\\right)^2}$$"
  },
  {
    "id": "Premaxilla_-_left",
    "name": "Premaxilla - left",
    "imageFile": "measurement_distance_left_premaxilla-lateral(l).png",
    "description": "Distance between 'Subnasale (sn)' and 'Alar Base Junction Left (ac_l)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Sn.x-Ac_l.x\\right)^2+\\left(Sn.y-Ac_l.y\\right)^2+\\left(Sn.z-Ac_l.z\\right)^2}$$"
  },
  {
    "id": "Premaxilla_-_right",
    "name": "Premaxilla - right",
    "imageFile": "measurement_distance_right_premaxilla-lateral(r).png",
    "description": "Distance between 'Subnasale (sn)' and 'Alar Base Junction Right (ac_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Sn.x-Ac_r.x\\right)^2+\\left(Sn.y-Ac_r.y\\right)^2+\\left(Sn.z-Ac_r.z\\right)^2}$$"
  },
  {
    "id": "Radix_Depth",
    "name": "Radix Depth",
    "imageFile": "measurement_distance_radix_depth-lateral(l).png",
    "description": "Distance between 'Nasal/Radix (n)' and 'Supratarsal Crease (sc_l)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Sc_l.x-N.x\\right)^2+\\left(Sc_l.y-N.y\\right)^2+\\left(Sc_l.z-N.z\\right)^2}$$"
  },
  {
    "id": "Radix_Projection",
    "name": "Radix Projection",
    "imageFile": "measurement_distance_radix_projection-lateral(l).png",
    "description": "Distance between 'Corneal Plane (cp)' and 'Nasion/Radix (n)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Cp.x-N.x\\right)^2+\\left(Cp.y-N.y\\right)^2+\\left(Cp.z-N.z\\right)^2}$$"
  },
  {
    "id": "Right_Eye_Width",
    "name": "Right Eye Width",
    "imageFile": "measurement_distance_right_eye_length-frontal.png",
    "description": "Distance from 'Endocanthion/Medial Canthus - right (en_r)' to 'Exocanthion/Lateral Canthus – right (ex_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(En_r.x-Ex_r.x\\right)^2+\\left(En_r.y-Ex_r.y\\right)^2+\\left(En_r.z-Ex_r.z\\right)^2}$$"
  },
  {
    "id": "Subnasal_Width",
    "name": "Subnasal Width",
    "imageFile": "measurement_distance_subnasal_width-basal.png",
    "description": "Distance between 'Subalare - right (sbal_r)' and 'Subalare - left (sbal_l)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Sbal_r.x-Sbal_l.x\\right)^2+\\left(Sbal_r.y-Sbal_l.y\\right)^2+\\left(Sbal_r.z-Sbal_l.z\\right)^2}$$"
  },
  {
    "id": "Supratip_Lobule",
    "name": "Supratip Lobule",
    "imageFile": "measurement_distance_supratip_lobule-lateral(l).png",
    "description": "Distance between 'Pronasale/Tip (prn)' and 'Supratip Break Point (s)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Prn.x-S.x\\right)^2+\\left(Prn.y-S.y\\right)^2+\\left(Prn.z-S.z\\right)^2}$$"
  },
  {
    "id": "Tip/Lobule_Width",
    "name": "Tip/Lobule Width",
    "imageFile": "measurement_distance_lobule_width-frontal.png",
    "description": "Distance from 'Tip Defining Point - left (td_l)' to 'Tip Defining Point - right (td_r)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Td_l.x-Td_r.x\\right)^2+\\left(Td_l.y-Td_r.y\\right)^2+\\left(Td_l.z-Td_r.z\\right)^2}$$"
  },
  {
    "id": "Upper_Facial_Height",
    "name": "Upper Facial Height",
    "imageFile": "measurement_distance_upper_facial_height-frontal.png",
    "description": "Distance between 'Trichion (tr)' and 'Glabella (g)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(G.x-Tr.x\\right)^2+\\left(G.y-Tr.y\\right)^2+\\left(G.z-Tr.z\\right)^2}$$"
  },
  {
    "id": "Upper_Lip_Height",
    "name": "Upper Lip Height",
    "imageFile": "measurement_distance_upper_lip_height-frontal.png",
    "description": "Distance between 'Subnasale (sn)' and 'Stomion (sto)'",
    "type": "Distances",
    "formula": "$$\\sqrt{\\left(Sn.x-Sto.x\\right)^2+\\left(Sn.y-Sto.y\\right)^2+\\left(Sn.z-Sto.z\\right)^2}$$"
  },
  {
    "id": "Columellar_Labial_Angle",
    "name": "Columellar Labial Angle",
    "imageFile": "measurement_angle_columellar_labial_angle-lateral(l).png",
    "description": " The columellar-labial angle is formed by the columella and the upper lip.",
    "type": "Angles",
    "formula": "V1=$$<\\left(Sn.x-Ls.x\\right),\\left(Sn.y-Ls.y\\right),\\left(Sn.z-Ls.z\\right)>$$ V2=$$<\\left(Sn.x-C.x\\right),\\left(Sn.y-C.y\\right),\\left(Sn.z-C.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Columellar_Lobular_Angle",
    "name": "Columellar Lobular Angle",
    "imageFile": "measurement_angle_columellar_lobular_angle-lateral(l).png",
    "description": "This angle is formed at the junction of the columella and the infratip lobule, and indicates the junction of the middle and medial crura.",
    "type": "Angles",
    "formula": "V1=$$<\\left(C.x-Prn.x\\right),\\left(C.y-Prn.y\\right),\\left(C.z-Prn.z\\right)>$$ V2=$$<\\left(C.x-Sn.x\\right),\\left(C.y-Sn.y\\right),\\left(C.z-Sn.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Degree_of_Deviation_(C-type)",
    "name": "Degree of Deviation (C-type)",
    "imageFile": "measurement_angle_degree_of_deviation_c_type-frontal.png",
    "description": "(A) glabella, (B) nasion, (C) nasal tip, (D) middle point of upper lip, (E) most prominent point of the convexity, C - type crooked nose angle: the angle between BE and EC lines.",
    "type": "Angles",
    "formula": "V1=$$<\\left(N.x-Pc.x\\right),\\left(N.y-Pc.y\\right),\\left(N.z-Pc.z\\right)>$$ V2=$$<\\left(Pc.x-Prn.x\\right),\\left(Pc.y-Prn.y\\right),\\left(Pc.z-Prn.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Degree_of_Deviation_(I-type)",
    "name": "Degree of Deviation (I-type)",
    "imageFile": "measurement_angle_degree_of_deviation_i_type-frontal.png",
    "description": "(A) glabella, (B) nasion, (C) most prominent point of the nasal tip, (D) middle point of upper lip, I - type crooked nose angle: the angle between BD and BC lines.",
    "type": "Angles",
    "formula": "V1=$$<\\left(N.x-Ls.x\\right),\\left(N.y-Ls.y\\right),\\left(N.z-Ls.z\\right)>$$ V2=$$<\\left(N.x-Prn.x\\right),\\left(N.y-Prn.y\\right),\\left(N.z-Prn.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Interalar_Angle",
    "name": "Interalar Angle",
    "imageFile": "measurement_angle_interalar_angle-basal.png",
    "description": "Angle between 'Alare - left (al_l)' to 'Pronasale/Tip (prn)' and 'Pronasale/Tip (prn)' to Alare - right (al_r))' Formed near the nose tip by the two tangents touching the alar on either side. Used to classify the nose shape into different morphological types.",
    "type": "Angles",
    "formula": "V1=$$<\\left(Al_l.x-Prn.x\\right),\\left(Al_l.y-Prn.y\\right),\\left(Al_l.z-Prn.z\\right)>$$ V2=$$<\\left(Prn.x-Al_r.x\\right),\\left(Prn.y-Al_r.y\\right),\\left(Prn.z-Al_r.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Interaxial_Angle_(Nostril_Inclination)",
    "name": "Interaxial Angle (Nostril Inclination)",
    "imageFile": "measurement_angle_nostril_inclination_with_midline-basal.png",
    "description": "Angle between 'Superior/Highest Point of Nostril - left (stn_l)' to 'Inferior/Lowest Point of Nostril - left (itn_l)' and 'Superior/Highest Point of Nostril - right (stn_r)' to 'Inferior/Lowest Point of Nostril - right (itn_r)' Formed at the apex where the two nostril axes meet; it is closely related to the nostril inclination",
    "type": "Angles",
    "formula": "V1=$$<\\left(Stn_l.x-Itn_l.x\\right),\\left(Stn_l.y-Itn_l.y\\right),\\left(Stn_l.z-Itn_l.z\\right)>$$ V2=$$<\\left(Stn_r.x-Itn_r.x\\right),\\left(Stn_r.y-Itn_r.y\\right),\\left(Stn_r.z-Itn_r.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Mentocervical_Angle",
    "name": "Mentocervical Angle",
    "imageFile": "measurement_angle_mentocervical_angle-lateral(l).png",
    "description": "The angle between 'Pronasale/Tip (prn)' to 'Pogonion (pg)' and 'Menton/Gnathion (me)' to 'Cervical Point (c)'",
    "type": "Angles",
    "formula": "V1=$$<\\left(Prn.x-Pg.x\\right),\\left(Prn.y-Pg.y\\right),\\left(Prn.z-Pg.z\\right)>$$ V2=$$<\\left(Me.x-C.x\\right),\\left(Me.y-C.y\\right),\\left(Me.z-C.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Nasal_Tip_Angle",
    "name": "Nasal Tip Angle",
    "imageFile": "measurement_angle_nasal_tip_angle-lateral(l).png",
    "description": " The angle between 'Nasion (n)' to 'Pronasale/Tip (prn)' and 'Pronasale/Tip (prn)' to 'Subnasale (sn)'",
    "type": "Angles",
    "formula": "V1=$$<\\left(N.x-Prn.x\\right),\\left(N.y-Prn.y\\right),\\left(N.z-Prn.z\\right)>$$ V2=$$<\\left(Prn.x-Sn.x\\right),\\left(Prn.y-Sn.y\\right),\\left(Prn.z-Sn.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Nasofacial_Angle",
    "name": "Nasofacial Angle",
    "imageFile": "measurement_angle_nasofacial_angle-lateral(l).png",
    "description": "The angle between 'Nasion (n)' to 'Tip/Pronasale (prn)' and 'Nasion (n)' to 'Pogonion (pg)'",
    "type": "Angles",
    "formula": "V1=$$<\\left(N.x-Pg.x\\right),\\left(N.y-Pg.y\\right),\\left(N.z-Pg.z\\right)>$$ V2=$$<\\left(N.x-Prn.x\\right),\\left(N.y-Prn.y\\right),\\left(N.z-Prn.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Nasofrontal_Angle",
    "name": "Nasofrontal Angle",
    "imageFile": "measurement_angle_nasofrontal_angle-lateral(l).png",
    "description": "The angle between 'Glabella (g)' to 'Nasion (n)' and 'Nasion (n)' to 'Pronasale/Tip (prn)'",
    "type": "Angles",
    "formula": "V1=$$<\\left(N.x-G.x\\right),\\left(N.y-G.y\\right),\\left(N.z-G.z\\right)>$$ V2=$$<\\left(N.x-G.x\\right),\\left(N.y-G.y\\right),\\left(N.z-G.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Nasolabial_Angle_1",
    "name": "Nasolabial Angle 1",
    "imageFile": "measurement_angle_nasolabial_angle-lateral(l).png",
    "description": "The angle between 'Pronasale/Tip (prn)' to 'Subnasale (sn)' and 'Subnasale (sn)' to 'Labrale Superius (ls)'",
    "type": "Angles",
    "formula": "V1=$$<\\left(Prn.x-Sn.x\\right),\\left(Prn.y-Sn.y\\right),\\left(Prn.z-Sn.z\\right)>$$ V2=$$<\\left(Sn.x-Ls.x\\right),\\left(Sn.y-Ls.y\\right),\\left(Sn.z-Ls.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Nasolabial_Angle_2",
    "name": "Nasolabial Angle 2",
    "imageFile": "measurement_angle_nasolabial_angle_II-lateral(r).png",
    "description": "The angle between the Columella and the line tangent to the cutaneous upper lip proper.",
    "type": "Angles",
    "formula": "$$$$"
  },
  {
    "id": "Nasolabial_Angle_3",
    "name": "Nasolabial Angle 3",
    "imageFile": "measurement_angle_nasolabial_angle_III-lateral(r).png",
    "description": "The angle between the long axis of the nostril and the line perpendicular to the Frankfort horizontal.",
    "type": "Angles",
    "formula": "$$$$"
  },
  {
    "id": "Nasolabial_Angle_4",
    "name": "Nasolabial Angle 4",
    "imageFile": "measurement_angle_nasolabial_angle_IV-lateral(r).png",
    "description": "The angle between the long axis of the nostril and the line perpendicular to the Frankfort horizontal.",
    "type": "Angles",
    "formula": "V1=$$<\\left(Itn_r.x-Itn_l.x\\right),\\left(Itn_r.y-Itn_l.y\\right),\\left(Itn_r.z-Itn_l.z\\right)>$$ V2=$$<\\left(G.x-Pg.x\\right),\\left(G.y-Pg.y\\right),\\left(G.z-Pg.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Nasomental_Angle",
    "name": "Nasomental Angle",
    "imageFile": "measurement_angle_nasomental_angle-lateral(l).png",
    "description": "The angle between 'Nasion (n)' to 'Pronasale/Tip (prn)' and 'Pronasale/Tip (prn)' to 'Pogonion (pg)'",
    "type": "Angles",
    "formula": "V1=$$<\\left(Prn.x-N.x\\right),\\left(Prn.y-N.y\\right),\\left(Prn.z-N.z\\right)>$$ V2=$$<\\left(Prn.x-Pg.x\\right),\\left(Prn.y-Pg.y\\right),\\left(Prn.z-Pg.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Total_Facial_Convexity_Angle",
    "name": "Total Facial Convexity Angle",
    "imageFile": "measurement_angle_total_facial_convexity_angle-lateral(l).png",
    "description": "Angle between 'Glabella (g)' to 'Pronasale/Tip (prn)' and 'Pronasale/Tip (prn)' to 'Pogonion (pg)'",
    "type": "Angles",
    "formula": "V1=$$<\\left(G.x-Prn.x\\right),\\left(G.y-Prn.y\\right),\\left(G.z-Prn.z\\right)>$$ V2=$$<\\left(Pg.x-Prn.x\\right),\\left(Pg.y-Prn.y\\right),\\left(Pg.z-Prn.z\\right)>$$$$\\theta=\\cos^{-1}(\\frac{V1\\bullet V2}{|V1||V2|})$$"
  },
  {
    "id": "Lobule_-_Base_Ratio",
    "name": "Lobule - Base Ratio",
    "imageFile": "measurement_ratio_lobule_base_ratio-basal.png",
    "description": "Calculated by dividing the width of the lobule by the width of the base of the nose.",
    "type": "Ratios",
    "formula": "$$\\frac{\\sqrt{\\left(Td_r.x-Td_l.x\\right)^2+\\left(Td_r.y-Td_l.y\\right)^2+\\left(Td_r.z-Td_l.z\\right)^2}}{\\sqrt{\\left(Ac_r.x-Ac_l.x\\right)^2+\\left(Ac_r.y-Ac_l.y\\right)^2+\\left(Ac_r.z-Ac_l.z\\right)^2}}$$"
  },
  {
    "id": "Nasal_Tip_Projection:_Baum",
    "name": "Nasal Tip Projection: Baum",
    "imageFile": "measurement_ratio_nasal_tip_projection_baum-lateral(l).png",
    "description": "The distance from the Nasion (n) to the point that creates a right triangle between with the Nasion (n) and the Pronasale (prn) divided by the distance from the Pronasale (prn) to this point.",
    "type": "Ratios",
    "formula": "BD=$$\\frac{|Prn-Sn|\\times|Prn-N|}{|N-Sn|}$$ AD=$$\\sqrt{\\left(N.x-Prn.x\\right)^2+\\left(N.y-Prn.y\\right)^2+\\left(N.z-Prn.z\\right)^2}$$ $$BD^2+AB^2=AD^2$$ AB=$$\\sqrt{AD^2-BD^2}$$ Projection=$$\\frac{AB}{BD}$$"
  },
  {
    "id": "Nasal_Tip_Projection:_Byrd_and_Hobar",
    "name": "Nasal Tip Projection: Byrd and Hobar",
    "imageFile": "measurement_ratio_nasal_tip_projection_byrd&hobar-lateral(l).png",
    "description": "The distance from the Nasion (n) to the Labiale Superius (ls) divided by the distance from the Pronasale (prn) to the Alar Crease (ac).",
    "type": "Ratios",
    "formula": "AT=$$\\sqrt{\\left(Prn.x-Ac_l.x\\right)^2+\\left(Prn.y-Ac_l.y\\right)^2+\\left(Prn.z-Ac_l.z\\right)^2}$$ 2A=$$\\frac{|Prn-Ls|\\times|Prn-N|}{|N-Ls|}$$ Projection=$$\\frac{2A}{AT}$$"
  },
  {
    "id": "Nasal_Tip_Projection:_Crumley_and_Lanser_I",
    "name": "Nasal Tip Projection: Crumley and Lanser I",
    "imageFile": "measurement_ratio_nasal_tip_projection_crumley&lanser1-lateral(l).png",
    "description": "The length of a line from the Nasion (n) to the Labiale Superius (ls) divided by the perpendicular distance from this line to the Pronasale (prn).",
    "type": "Ratios",
    "formula": "AE=$$\\sqrt{\\left(N.x-Ls.x\\right)^2+\\left(N.y-Ls.y\\right)^2+\\left(N.z-Ls.z\\right)^2}$$ BD=$$\\frac{|Prn-Ls|\\times|Prn-N|}{|N-Ls|}$$ Projection=$$\\frac{AE}{BD}$$"
  },
  {
    "id": "Nasal_Tip_Projection:_Crumley_and_Lanser_II",
    "name": "Nasal Tip Projection: Crumley and Lanser II",
    "imageFile": "measurement_ratio_nasal_tip_projection_crumley&lanser2-lateral(l).png",
    "description": "The length of a line from the Nasion (n) to the Menton/Gnathion (me) divided by the perpendicular distance from the Pronasale (prn) to this line.",
    "type": "Ratios",
    "formula": "AZ=$$\\sqrt{\\left(N.x-Me.x\\right)^2+\\left(N.y-Me.y\\right)^2+\\left(N.z-Me.z\\right)^2}$$ YD=$$\\frac{|Prn-Me|\\times|Prn-N|}{|N-Me|}$$ Projection=$$\\frac{AZ}{YD}$$"
  },
  {
    "id": "Nasal_Tip_Projection:_Goode",
    "name": "Nasal Tip Projection: Goode",
    "imageFile": "measurement_ratio_nasal_tip_projection_goode-lateral(l).png",
    "description": "The distance from Pronasale  (prn) to Alar Base Junction/Alar Crease - right (ac r) divided by the distance from Nasion (n) to Nasal Tip/Pronasale (prn).",
    "type": "Ratios",
    "formula": "$$\\frac{\\sqrt{\\left(Ac_l.x-Prn.x\\right)^2+\\left(Ac_l.y-Prn.y\\right)^2+\\left(Ac_l.z-Prn.z\\right)^2}}{\\sqrt{\\left(Prn.x-N.x\\right)^2+\\left(Prn.y-N.y\\right)^2+\\left(Prn.z-N.z\\right)^2}}$$"
  },
  {
    "id": "Nasal_Tip_Projection:_Powell_and_Humphries",
    "name": "Nasal Tip Projection: Powell and Humphries",
    "imageFile": "measurement_ratio_nasal_tip_projection_powell&humphries-lateral(l).png",
    "description": "The length of a line from Subnasale (sn) to Nasion (n) divided by the perpendicular distance from this line to Nasal Tip/Pronasale (prn).",
    "type": "Ratios",
    "formula": "AC=$$\\sqrt{\\left(N.x-Sn.x\\right)^2+\\left(N.y-Sn.y\\right)^2+\\left(N.z-Sn.z\\right)^2}$$ BD=$$\\frac{|Prn-Sn|\\times|Prn-N|}{|N-Sn|}$$ Projection=$$\\frac{AC}{BD}$$"
  },
  {
    "id": "Nasal_Tip_Projection:_Simons",
    "name": "Nasal Tip Projection: Simons",
    "imageFile": "measurement_ratio_nasal_tip_projection_simons-lateral(l).png",
    "description": "The distance from Subnasale (sn) to Nasal Tip/Pronasale (prn) divided by the distance from Subnasale (sn) to Labiale Superius (ls).",
    "type": "Ratios",
    "formula": "$$\\frac{\\sqrt{\\left(Sn.x-Prn.x\\right)^2+\\left(Sn.y-Prn.y\\right)^2+\\left(Sn.z-Prn.z\\right)^2}}{\\sqrt{\\left(Sn.x-Ls.x\\right)^2+\\left(Sn.y-Ls.y\\right)^2+\\left(Sn.z-Ls.z\\right)^2}}$$"
  },
  {
    "id": "Nasal_Width–Intercanthal_Distance_Ratio",
    "name": "Nasal Width–Intercanthal Distance Ratio",
    "imageFile": "measurement_ratio_nasal_width_intercanthal_distance_ratio-frontal.png",
    "description": "Nasal Width (interalar distance) divided by intercanthal distance.",
    "type": "Ratios",
    "formula": "$$\\frac{Interalar Distance}{(Inner) Intercanthal Distance}$$ $$\\frac{\\sqrt{\\left(Al_r.x-Al_l.x\\right)^2+\\left(Al_r.y-Al_l.y\\right)^2+\\left(Al_r.z-Al_l.z\\right)^2}}{\\sqrt{\\left(En_r.x-En_l.x\\right)^2+\\left(En_r.y-En_l.y\\right)^2+\\left(En_r.z-En_l.z\\right)^2}}$$"
  },
  {
    "id": "Nasal_Width-Length_Ratio",
    "name": "Nasal Width-Length Ratio",
    "imageFile": "measurement_ratio_nasal_width_length_ratio-frontal.png",
    "description": "The alar width divided by the nasal bridge length. Nasal width (ala to ala) divided by the length of the nose (nasion to pronasion)",
    "type": "Ratios",
    "formula": "$$\\frac{Interalar Distance}{Nasal Bridge Length}$$ $$\\frac{\\sqrt{\\left(Al_r.x-Al_l.x\\right)^2+\\left(Al_r.y-Al_l.y\\right)^2+\\left(Al_r.z-Al_l.z\\right)^2}}{\\sqrt{\\left(Prn.x-N.x\\right)^2+\\left(Prn.y-N.y\\right)^2+\\left(Prn.z-N.z\\right)^2}}$$"
  }
]