4_CRD )_PUR REMARKS_NN2 ON_PRP CIRCULAR_NN1 HORIZON_NN1 ARCS_NN2 
It_PNP should_VM0 be_VBI plainly_AV0 evident_AJ0 at_PRP this_DT0 point_NN1 that_CJT horizon_NN1 arcs_NN2 are_VBB not_XX0 generally_AV0 circular_AJ0 —_PUN even_AV0 for_PRP a_AT0 spherical_AJ0 celestial_AJ0 body_NN1 ._SENT Indeed_AV0 ,_PUN we_PNP have_VHB already_AV0 shown_VVN that_CJT spherical_AJ0 bodies_NN2 may_VM0 produce_VVI hyperbolic_AJ0 horizon_NN1 arcs_NN2 when_CJS imaged_VVN from_PRP a_AT0 low_AJ0 altitude_NN1 ._SENT Despite_PRP this_DT0 ,_PUN there_EX0 seems_VVZ to_TO0 be_VBI a_AT0 widespread_AJ0 misconception_NN1 that_CJT assuming_VVG circular_AJ0 horizon_NN1 arcs_NN2 for_PRP OPNAV_NP0 is_VBZ a_AT0 good_AJ0 approximation_NN1 and_CJC that_CJT such_DT0 an_AT0 approach_NN1 has_VHZ the_AT0 benefit_NN1 of_PRF '_PUQ '_POS simplicity_NN1 ._SENT '_PUQ '_PUQ Both_DT0 claims_NN2 are_VBB almost_AV0 always_AV0 false_AJ0 ._SENT The_AT0 results_NN2 of_PRF this_DT0 tutorial_NN1 clearly_AV0 show_VVB there_EX0 is_VBZ no_AT0 justification_NN1 for_PRP the_AT0 modern_AJ0 OPNAV_NN1 practitioner_NN1 to_TO0 ever_AV0 presuppose_VVI a_AT0 circular_AJ0 horizon_NN1 arc_NN1 ._SENT 
We_PNP begin_VVB our_DPS remarks_NN2 by_PRP acknowledging_VVG that_CJT spherical_AJ0 bodies_NN2 positioned_VVN along_PRP the_AT0 camera_NN1 boresight_NN1 will_VM0 produce_VVI a_AT0 circular_AJ0 horizon_NN1 arc_NN1 ._SENT This_DT0 may_VM0 be_VBI intuitively_AV0 deduced_VVN by_PRP taking_VVG a_AT0 slice_NN1 of_PRF a_AT0 right_AJ0 circular_AJ0 cone_NN1 with_PRP an_AT0 image_NN1 plane_NN1 perpendicular_NN1 the_AT0 cone_NN1 's_POS axis_NN1 of_PRF revolution_NN1 (_PUL which_DTQ ,_PUN in_PRP the_AT0 special_AJ0 case_NN1 of_PRF a_AT0 sphere_NN1 ,_PUN is_VBZ also_AV0 the_AT0 direction_NN1 from_PRP the_AT0 camera_NN1 to_PRP the_AT0 body_NN1 center_NN1 )_PUR ._SENT We_PNP may_VM0 also_AV0 arrive_VVI at_PRP this_DT0 result_NN1 algebraically_AV0 by_PRP recalling_VVG that_CJT the_AT0 projected_AJ0 conic_AJ0 is_VBZ a_AT0 circle_NN1 when_AVQ the_AT0 coefficients_NN2 of_PRF the_AT0 implicit_AJ0 conic_AJ0 equation_NN1 obey_VVB the_AT0 constraint_NN1 (_PUL A_ZZ0 −_NN1 C_ZZ0 )_PUR 2_CRD +_NP0 B2_CRD =_NP0 0._CRD For_PRP a_AT0 spherical_AJ0 body_NN1 ,_PUN substitution_NN1 from_PRP Eq._NP0 (_PUL 65_CRD )_PUR shows_VVZ that_CJT equation_NN1 (_PUL 75_CRD )_PUR This_DT0 implies_VVZ that_CJT XC_CRD =_NP0 0_CRD and_CJC YC_NP0 =_NP0 0_CRD ,_PUN which_DTQ means_VVZ the_AT0 vector_NN1 from_PRP the_AT0 camera_NN1 to_PRP the_AT0 celestial_AJ0 body_NN1 center_NN1 is_VBZ rTC_NP0 =_NP0 [_PUL 0_CRD ,_PUN 0_CRD ,_PUN ZC_NP0 ]_PUR and_CJC is_VBZ entirety_NN1 along_PRP the_AT0 camera_NN1 boresight_NN1 direction_NN1 ._SENT Whenever_AVQ the_AT0 camera_NN1 boresight_NN1 is_VBZ not_XX0 pointed_VVN exactly_AV0 at_PRP the_AT0 body_NN1 's_POS center_NN1 (_PUL for_PRP a_AT0 sphere_NN1 )_PUR ,_PUN the_AT0 projected_AJ0 horizon_NN1 arc_NN1 will_VM0 be_VBI something_PNI other_AJ0 than_CJS a_AT0 circle_NN1 (_PUL e.g._AV0 ,_PUN an_AT0 ellipse_NN1 )_PUR ._SENT 
For_PRP a_AT0 spherical_AJ0 body_NN1 having_VHG ZC_NP0 >_NP0 a_AT0 ,_PUN the_AT0 conic_AJ0 locus_NN1 of_PRF the_AT0 horizon_NN1 is_VBZ an_AT0 ellipse_NN1 given_VVN by_PRP Eq._NP0 (_PUL 64_CRD )_PUR ._SENT Some_DT0 tedious_AJ0 algebra_NN1 will_VM0 show_VVI this_DT0 horizon_NN1 arc_VVB to_TO0 have_VHI an_AT0 ellipticity_NN1 (_PUL ε_NP0 =_NP0 ah_ITJ /_PUN bh_ITJ ;_PUN ratio_NN1 of_PRF semimajor_NN1 ah_ITJ to_PRP semiminor_NN1 axis_NN1 bh_NN1 in_PRP image_NN1 )_PUR of_PRF equation_NN1 (_PUL 76_CRD )_PUR which_DTQ produces_VVZ an_AT0 eccentricity_NN1 e_ZZ0 of_PRF equation_NN1 (_PUL 77_CRD )_PUR where_CJS φ_NN1 is_VBZ the_AT0 half-angle_NN1 of_PRF the_AT0 tightly_AV0 bounding_AJ0 cone_NN1 and_CJC ψ_NN1 is_VBZ the_AT0 angle_NN1 between_PRP rC_NN1 and_CJC the_AT0 camera_NN1 boresight_NN1 direction_NN1 ._SENT These_DT0 angle_NN1 definitions_NN2 are_VBB the_AT0 same_DT0 as_CJS illustrated_VVN in_PRP Fig._NN1 13._CRD The_AT0 ellipticity_NN1 is_VBZ often_AV0 more_AV0 extreme_AJ0 for_PRP non-spherical_AJ0 bodies_NN2 ._SENT Moreover_AV0 ,_PUN we_PNP find_VVB the_AT0 ellipticity_NN1 of_PRF the_AT0 projected_AJ0 horizon_NN1 (_PUL even_AV0 if_CJS small_AJ0 )_PUR to_TO0 be_VBI critical_AJ0 for_PRP navigation_NN1 performance_NN1 and_CJC solution_NN1 stability_NN1 ._SENT The_AT0 navigation_NN1 analyst_NN1 ignores_VVZ this_DT0 effect_NN1 at_PRP their_DPS own_DT0 peril_NN1 ._SENT 
F_ZZ0 ._SENT HORIZON_NN1 LOCALIZATION_NN1 IN_PRP A_AT0 DIGITAL_AJ0 IMAGE_NN1 
For_PRP an_AT0 image_NN1 of_PRF a_AT0 celestial_AJ0 body_NN1 to_TO0 be_VBI useful_AJ0 for_PRP navigation_NN1 ,_PUN we_PNP must_VM0 be_VBI capable_AJ0 of_PRF automatically_AV0 extracting_VVG the_AT0 pixel_NN1 coordinates_NN2 of_PRF points_NN2 of_PRF interest_NN1 ._SENT In_PRP this_DT0 work_NN1 ,_PUN we_PNP are_VBB interested_AJ0 in_PRP points_NN2 belonging_VVG to_PRP the_AT0 body_NN1 's_POS apparent_AJ0 horizon_NN1 ._SENT Finding_VVG these_DT0 points_NN2 is_VBZ an_AT0 image_NN1 processing_NN1 task_NN1 ._SENT 
Less_AV0 than_CJS half_DT0 of_PRF the_AT0 apparent_AJ0 horizon_NN1 is_VBZ usable_AJ0 for_PRP navigation_NN1 in_PRP a_AT0 typical_AJ0 image_NN1 ._SENT Except_CJS in_PRP rare_AJ0 special_AJ0 cases_NN2 ,_PUN images_NN2 of_PRF celestial_AJ0 bodies_NN2 will_VM0 be_VBI at_PRP non-zero_AJ0 phase_NN1 angles_NN2 and_CJC the_AT0 terminator_NN1 (_PUL day_NN1 /_PUN night_NN1 line_NN1 )_PUR will_VM0 be_VBI visible_AJ0 ._SENT As_CJS a_AT0 consequence_NN1 ,_PUN part_NN1 of_PRF the_AT0 horizon_NN1 will_VM0 be_VBI illuminated_VVN (_PUL we_PNP call_VVB this_DT0 the_AT0 lit_NN1 limb_NN1 )_PUR and_CJC part_NN1 will_VM0 be_VBI in_PRP shadow_NN1 ._SENT Only_AV0 the_AT0 lit_NN1 limb_NN1 is_VBZ useful_AJ0 for_PRP navigation_NN1 ._SENT A_AT0 review_NN1 of_PRF the_AT0 example_NN1 images_NN2 from_PRP Fig._NN1 8_CRD will_VM0 show_VVI the_AT0 shadowed_AJ0 limb_NN1 to_TO0 be_VBI nearly_AV0 undetectable_AJ0 (_PUL when_CJS it_PNP is_VBZ detectable_AJ0 ,_PUN the_AT0 signal-to-noise_AJ0 ratio_NN1 is_VBZ poor_AJ0 )_PUR ._SENT Moreover_AV0 ,_PUN we_PNP generally_AV0 avoid_VVB using_VVG points_NN2 along_PRP the_AT0 lit_NN1 limb_NN1 near_PRP the_AT0 cusps_NN2 ,_PUN where_CJS the_AT0 shadow_NN1 of_PRF the_AT0 terminator_NN1 begins_VVZ to_TO0 encroach_VVI on_PRP the_AT0 lit_NN1 horizon_NN1 —_PUN the_AT0 result_NN1 is_VBZ less_AV0 than_CJS 180_CRD deg_NN0 of_PRF usable_AJ0 horizon_NN1 arc_NN1 in_PRP almost_AV0 all_DT0 practical_AJ0 cases_NN2 ._SENT Past_AJ0 studies_NN2 show_VVB the_AT0 best_AJS localization_NN1 performance_NN1 is_VBZ about_PRP ±_NN0 70_CRD deg_NN0 from_PRP the_AT0 limb_NN1 's_POS subsolar_AJ0 point_NN1 (_PUL for_PRP a_AT0 total_NN1 of_PRF 140_CRD deg_NN0 of_PRF horizon_NN1 arc_NN1 )_PUR ._SENT 
Before_AV0 proceeding_VVG further_AJC ,_PUN the_AT0 reader_NN1 is_VBZ explicitly_AV0 warned_VVN against_PRP using_VVG the_AT0 entire_AJ0 lit_NN1 limb_NN1 (_PUL 360_CRD deg_NN0 of_PRF arc_NN1 )_PUR at_PRP very_AV0 low_AJ0 phase_NN1 angle_NN1 ,_PUN even_AV0 though_CJS the_AT0 celestial_AJ0 body_NN1 's_POS disk_NN1 may_VM0 qualitatively_AV0 appear_VVI to_TO0 be_VBI fully_AV0 illuminated_VVN ._SENT In_PRP these_DT0 cases_NN2 ,_PUN it_PNP is_VBZ not_XX0 uncommon_AJ0 for_PRP the_AT0 terminator_NN1 to_PRP lightly_AV0 shadow_NN1 (_PUL or_CJC slightly_AV0 trim_VVB a_AT0 few_DT0 pixels_NN2 off_AVP )_PUR one_CRD side_NN1 of_PRF the_AT0 apparent_AJ0 disk_NN1 —_PUN even_AV0 if_CJS it_PNP is_VBZ indiscernible_AJ0 from_PRP casual_AJ0 inspection_NN1 by_PRP the_AT0 human_AJ0 analyst_NN1 ._SENT The_AT0 shadowed_AJ0 (_PUL or_CJC trimmed_AJ0 )_PUR horizon_NN1 pixels_NN2 will_VM0 shift_VVI the_AT0 apparent_AJ0 limb_NN1 location_NN1 towards_PRP the_AT0 center_NN1 of_PRF the_AT0 body_NN1 and_CJC corrupt_AJ0 the_AT0 resulting_AJ0 navigation_NN1 solution_NN1 in_PRP a_AT0 meaningful_AJ0 way_NN1 ._SENT Outside_AV0 of_PRF a_AT0 simulation_NN1 environment_NN1 ,_PUN superior_AJ0 performance_NN1 is_VBZ almost_AV0 always_AV0 achieved_VVN by_PRP restricting_VVG ourselves_PNX to_PRP only_AV0 using_VVG horizon_NN1 points_NN2 belonging_VVG to_PRP the_AT0 lit_NN1 limb_NN1 ._SENT 
For_PRP an_AT0 airless_AJ0 body_NN1 (_PUL e.g._AV0 ,_PUN Moon_NP0 ,_PUN Pluto_NP0 )_PUR ,_PUN there_EX0 is_VBZ a_AT0 discontinuity_NN1 in_PRP brightness_NN1 across_PRP the_AT0 lit_NN1 limb_NN1 ._SENT This_DT0 discontinuity_NN1 is_VBZ spread_VVN out_AVP in_PRP a_AT0 digital_AJ0 image_NN1 due_AJ0 to_PRP the_AT0 the_AT0 optical_AJ0 assembly_NN1 's_POS point_NN1 spread_NN1 function_NN1 and_CJC spatial_AJ0 quantization_NN1 by_PRP the_AT0 FPA_NN1 detector_NN1 ._SENT Regardless_AV0 ,_PUN the_AT0 underlying_AJ0 brightness_NN1 discontinuity_NN1 makes_VVZ limb_NN1 localization_NN1 for_PRP an_AT0 airless_AJ0 body_NN1 equivalent_NN1 to_PRP an_AT0 edge_NN1 localization_NN1 problem_NN1 ._SENT There_EX0 are_VBB a_AT0 variety_NN1 of_PRF widely-available_AJ0 edge_NN1 localization_NN1 algorithms_NN2 with_PRP pixel-level_AJ0 performance_NN1 (_PUL i.e._AV0 ,_PUN these_DT0 algorithms_NN2 identify_VVB edge_NN1 pixels_NN2 )_PUR ,_PUN including_VVG :_PUN Sobel_NP0 ,_PUN Marr-Hilderth_NP0 ,_PUN Canny_AJ0 ,_PUN and_CJC many_DT0 others_NN2 ._SENT These_DT0 pixel-level_AJ0 methods_NN2 usually_AV0 provide_VVB reasonable_AJ0 qualitative_AJ0 performance_NN1 —_PUN making_VVG it_PNP possible_AJ0 to_TO0 produce_VVI navigation_NN1 solutions_NN2 ,_PUN compute_VVB statistics_NN0 ,_PUN and_CJC write_VVB papers_NN2 ._SENT In_PRP practice_NN1 ,_PUN however_AV0 ,_PUN these_DT0 pixel-level_AJ0 edge_NN1 locations_NN2 are_VBB inadequate_AJ0 for_PRP navigation_NN1 ._SENT Horizon-based_AJ0 navigation_NN1 using_VVG fits_NN2 to_PRP image_NN1 limb_NN1 points_NN2 absolutely_AV0 requires_VVZ subpixel_AJ0 horizon_NN1 localization_NN1 with_PRP contemporary_AJ0 (_PUL circa_AV0 2020_CRD )_PUR imaging_NN1 sensors_NN2 ._SENT Good_AJ0 results_NN2 are_VBB achievable_AJ0 with_PRP the_AT0 partial_AJ0 area_NN1 effect_NN1 (_PUL PAE_NP0 )_PUR technique_NN1 and_CJC best-in-class_AJ0 performance_NN1 is_VBZ achievable_AJ0 with_PRP moment-based_AJ0 methods_NN2 ._SENT The_AT0 reader_NN1 is_VBZ directed_VVN to_PRP the_AT0 image_NN1 processing_NN1 literature_NN1 for_PRP additional_AJ0 details_NN2 on_PRP the_AT0 edge_NN1 localization_NN1 problem_NN1 ._SENT 
Regardless_AV0 of_PRF the_AT0 algorithm_NN1 chosen_VVN ,_PUN the_AT0 end_NN1 result_NN1 is_VBZ usually_AV0 a_AT0 set_NN1 of_PRF n_ZZ0 pixel_NN1 coordinates_NN2 along_PRP the_AT0 celestial_AJ0 body_NN1 's_VHZ lit_VVN limb_NN1 ._SENT We_PNP can_VM0 describe_VVI this_DT0 set_NN1 of_PRF points_NN2 in_PRP homogeneous_AJ0 coordinates_NN2 as_CJS {_PUL u_ZZ0 ̄i_AJ0 }_PUR ni_NN2 =_NN1 1._CRD 
Finally_AV0 ,_PUN the_AT0 reader_NN1 is_VBZ cautioned_VVN against_PRP making_VVG undue_AJ0 use_NN1 of_PRF the_AT0 apparent_AJ0 terminator_NN1 location_NN1 ._SENT It_PNP is_VBZ true_AJ0 that_CJT the_AT0 terminator_NN1 provides_VVZ a_AT0 direct_AJ0 measurement_NN1 of_PRF phase_NN1 angle_NN1 ,_PUN sometimes_AV0 making_VVG it_PNP useful_AJ0 as_PRP a_AT0 means_NN0 of_PRF determining_VVG the_AT0 direction_NN1 to_PRP the_AT0 Sun_NN1 ._SENT However_AV0 ,_PUN because_CJS the_AT0 terminator_NN1 arc_NN1 occurs_VVZ where_AVQ the_AT0 incoming_AJ0 sunlight_NN1 is_VBZ tangent_AJ0 to_PRP the_AT0 body_NN1 's_POS ellipsoidal_AJ0 surface_NN1 ,_PUN the_AT0 sensitivity_NN1 of_PRF terminator_NN1 location_NN1 to_PRP sunlight_NN1 direction_NN1 is_VBZ quite_AV0 poor_AJ0 ._SENT Further_AV0 ,_PUN unlike_PRP the_AT0 lit_NN1 limb_NN1 ,_PUN the_AT0 terminator_NN1 does_VDZ not_XX0 correspond_VVI to_PRP a_AT0 physical_AJ0 discontinuity_NN1 that_CJT manifests_VVZ as_PRP an_AT0 underlying_AJ0 discontinuity_NN1 in_PRP image_NN1 intensity_NN1 ._SENT Instead_AV0 ,_PUN the_AT0 terminator_NN1 arc_NN1 is_VBZ a_AT0 locus_NN1 of_PRF points_NN2 on_PRP the_AT0 interior_NN1 of_PRF the_AT0 celestial_AJ0 body_NN1 's_POS apparent_AJ0 disk_NN1 ._SENT Accurate_AJ0 localization_NN1 of_PRF the_AT0 terminator_NN1 in_PRP an_AT0 image_NN1 is_VBZ difficult_AJ0 because_CJS the_AT0 transition_NN1 from_PRP sunlight_NN1 to_PRP shadow_NN1 is_VBZ gradual_AJ0 and_CJC often_AV0 occurs_VVZ over_PRP a_AT0 very_AV0 large_AJ0 number_NN1 of_PRF pixels_NN2 ._SENT Moreover_AV0 ,_PUN real_AJ0 bodies_NN2 have_VHB texture_NN1 that_CJT further_AV0 complicates_VVZ the_AT0 fitting_NN1 of_PRF simple_AJ0 models_NN2 to_PRP the_AT0 apparent_AJ0 terminator_NN1 intensity_NN1 gradient_NN1 —_PUN specifically_AV0 (_PUL 1_CRD )_PUR globally_AV0 varying_AJ0 surface_NN1 albedo_NN1 and_CJC (_PUL 2_CRD )_PUR terrain_NN1 relief_NN1 that_CJT casts_VVZ long_AJ0 shadows_NN2 near_PRP the_AT0 terminator_NN1 ._SENT Until_CJS new_AJ0 techniques_NN2 resolve_VVB these_DT0 challenges_NN2 ,_PUN the_AT0 combination_NN1 of_PRF poor_AJ0 sensitivity_NN1 to_PRP incoming_AJ0 sunlight_NN1 direction_NN1 and_CJC poor_AJ0 localization_NN1 in_PRP real_AJ0 imagery_NN1 makes_VVZ the_AT0 terminator_NN1 a_AT0 poor_AJ0 navigation_NN1 observable_AJ0 for_PRP most_DT0 applications_NN2 ._SENT 
G_ZZ0 ._SENT IMAGING_AJ0 CELESTIAL_AJ0 BODIES_NN2 AT_PRP DIFFERENT_AJ0 WAVELENGTHS_NN2 
Most_DT0 space_NN1 imaging_NN1 systems_NN2 used_VVN for_PRP navigation_NN1 operate_VVB in_PRP either_AV0 the_AT0 visible_AJ0 or_CJC infrared_AJ0 spectrum_NN1 ._SENT It_PNP is_VBZ often_AV0 preferable_AJ0 to_TO0 work_VVI in_PRP the_AT0 visible_AJ0 spectrum_NN1 since_PRP these_DT0 cameras_NN2 have_VHB higher_AJC resolution_NN1 FPAs_NN2 ,_PUN produce_VVB less_AV0 noisy_AJ0 images_NN2 ,_PUN are_VBB cheaper_AJC ,_PUN and_CJC place_VVB fewer_DT0 engineering_NN1 requirements_NN2 on_PRP the_AT0 vehicle_NN1 (_PUL e.g._AV0 ,_PUN thermal_AJ0 control_NN1 )_PUR ._SENT Observations_NN2 in_PRP the_AT0 visible_AJ0 spectrum_NN1 work_NN1 exceptionally_AV0 well_AV0 on_PRP airless_AJ0 bodies_NN2 where_AVQ there_EX0 is_VBZ a_AT0 crisp_AJ0 and_CJC predictable_AJ0 horizon_NN1 line_NN1 ._SENT 
For_PRP a_AT0 celestial_AJ0 body_NN1 with_PRP an_AT0 atmosphere_NN1 (_PUL e.g._AV0 ,_PUN Earth_NN1 ,_PUN Venus_NP0 )_PUR ,_PUN there_EX0 is_VBZ not_XX0 a_AT0 brightness_NN1 discontinuity_NN1 across_PRP the_AT0 lit_NN1 limb_NN1 when_CJS imaged_VVN in_PRP the_AT0 visible_AJ0 spectrum_NN1 ._SENT The_AT0 scattering_NN1 of_PRF sunlight_NN1 in_PRP the_AT0 atmosphere_NN1 causes_VVZ a_AT0 gradual_AJ0 change_NN1 in_PRP image_NN1 brightness_NN1 spanning_VVG the_AT0 width_NN1 of_PRF the_AT0 atmosphere_NN1 ._SENT Thus_AV0 ,_PUN in_PRP the_AT0 visible_AJ0 spectrum_NN1 ,_PUN we_PNP find_VVB that_CJT the_AT0 atmosphere_NN1 obscures_VVZ the_AT0 true_AJ0 horizon_NN1 and_CJC creates_VVZ an_AT0 apparent_AJ0 horizon_NN1 at_PRP an_AT0 unpredictable_AJ0 altitude_NN1 above_PRP the_AT0 surface_NN1 ._SENT This_DT0 is_VBZ a_AT0 long-standing_AJ0 challenge_NN1 for_PRP horizon-based_AJ0 OPNAV_NP0 and_CJC the_AT0 specific_AJ0 issues_NN2 are_VBB addressed_VVN at_PRP length_NN1 elsewhere_AV0 ._SENT The_AT0 usual_AJ0 solution_NN1 for_PRP robotic_AJ0 space_NN1 exploration_NN1 has_VHZ been_VBN to_TO0 ignore_VVI bodies_NN2 with_PRP atmospheres_NN2 and_CJC to_TO0 only_AV0 use_VVI images_NN2 of_PRF airless_AJ0 bodies_NN2 for_PRP horizon-based_AJ0 OPNAV_NP0 ,_PUN which_DTQ has_VHZ been_VBN satisfactory_AJ0 for_PRP past_AJ0 missions_NN2 ._SENT While_CJS there_EX0 has_VHZ been_VBN some_DT0 very_AV0 preliminary_AJ0 studies_NN2 of_PRF overcoming_VVG this_DT0 problem_NN1 for_PRP autonomous_AJ0 OPNAV_NP0 ,_PUN this_DT0 early_AJ0 work_NN1 is_VBZ incomplete_AJ0 and_CJC suggests_VVZ no_AT0 specific_AJ0 path_NN1 forward_AV0 ._SENT This_DT0 is_VBZ an_AT0 area_NN1 ripe_AJ0 for_PRP future_AJ0 research_NN1 ._SENT 
Past_AJ0 spaceflight_NN1 missions_NN2 have_VHB not_XX0 made_VVN extensive_AJ0 use_NN1 of_PRF infrared_AJ0 images_NN2 for_PRP horizon-based_AJ0 navigation_NN1 because_CJS of_PRF (_PUL 1_CRD )_PUR their_DPS lack_NN1 of_PRF availability_NN1 and_CJC (_PUL 2_CRD )_PUR poor_AJ0 performance_NN1 stemming_VVG from_PRP their_DPS comparatively_AV0 low_AJ0 FPA_NN1 resolution_NN1 ._SENT For_PRP most_DT0 planetary_AJ0 exploration_NN1 applications_NN2 —_PUN especially_AV0 to_PRP airless_AJ0 bodies_NN2 —_PUN visible_AJ0 spectrum_NN1 images_NN2 are_VBB preferred_VVN ._SENT Conversely_AV0 ,_PUN when_CJS we_PNP must_VM0 use_VVI images_NN2 of_PRF Earth_NN1 ,_PUN sensing_VVG in_PRP the_AT0 infrared_NN1 becomes_VVZ a_AT0 more_AV0 attractive_AJ0 option_NN1 ._SENT The_AT0 Earth_NN1 's_POS atmosphere_NN1 is_VBZ known_VVN to_TO0 possess_VVI a_AT0 crisp_AJ0 horizon_NN1 in_PRP the_AT0 longwave_NN1 infrared_NN1 (_PUL LWIR_NP0 )_PUR at_PRP wavelengths_NN2 of_PRF 14.0_CRD –_PUN 16.3_CRD μm_NN2 ._SENT This_DT0 is_VBZ due_AJ0 to_PRP infrared_AJ0 radiation_NN1 emitted_VVN from_PRP the_AT0 atmosphere_NN1 's_POS CO2_AJ0 band_NN1 ,_PUN which_DTQ produces_VVZ an_AT0 apparent_AJ0 horizon_NN1 at_PRP an_AT0 altitude_NN1 of_PRF about_PRP 30_CRD –_PUN 50_CRD km_NN0 ._SENT For_PRP these_DT0 reasons_NN2 ,_PUN most_DT0 Earth_NN1 horizon_NN1 sensors_NN2 used_VVN for_PRP attitude_NN1 determination_NN1 operate_VVB in_PRP the_AT0 LWIR_NN1 ._SENT 
H_ZZ0 ._SENT REMARKS_NN2 ON_PRP CONIC_AJ0 FITTING_NN1 IN_PRP AN_AT0 IMAGE_NN1 
Points_NN2 lying_VVG on_PRP an_AT0 ellipsoidal_AJ0 body_NN1 's_POS horizon_NN1 form_NN1 a_AT0 conic_AJ0 in_PRP an_AT0 image_NN1 (_PUL see_VVB Sec._NN1 II-D_NN1 )_PUR ._SENT Thus_AV0 ,_PUN if_CJS we_PNP are_VBB able_AJ0 to_TO0 find_VVI horizon_NN1 points_NN2 in_PRP an_AT0 image_NN1 (_PUL see_VVB Sec._NN1 II-F_NN1 )_PUR ,_PUN it_PNP is_VBZ sometimes_AV0 useful_AJ0 to_TO0 fit_VVI a_AT0 conic_AJ0 to_PRP these_DT0 measured_AJ0 horizon_NN1 points_NN2 ._SENT An_AT0 illustrative_AJ0 example_NN1 is_VBZ shown_VVN in_PRP Fig._NN1 14._CRD 
Horizon_NN1 arcs_NN2 are_VBB usually_AV0 elliptical_AJ0 or_CJC hyperbolic_AJ0 (_PUL see_VVB Sec._NN1 II-E3_CRD )_PUR ,_PUN making_VVG circle_NN1 fitting_NN1 techniques_NN2 (_PUL e.g._AV0 ,_PUN Taubin_NP0 's_POS method_NN1 )_PUR inappropriate_AJ0 for_PRP the_AT0 present_AJ0 application_NN1 ._SENT In_PRP the_AT0 case_NN1 where_AVQ the_AT0 spacecraft_NN0 is_VBZ always_AV0 far_AV0 away_AV0 from_PRP the_AT0 observed_AJ0 celestial_AJ0 body_NN1 ,_PUN the_AT0 horizon_NN1 will_VM0 always_AV0 be_VBI an_AT0 ellipse_NN1 (_PUL e.g._AV0 ,_PUN see_VVB Fig._NN1 13_CRD )_PUR and_CJC ellipse-specific_AJ0 methods_NN2 may_VM0 be_VBI used_VVN in_PRP these_DT0 situations_NN2 ._SENT Otherwise_AV0 ,_PUN it_PNP is_VBZ better_AJC to_TO0 use_VVI generic_AJ0 conic_AJ0 fitting_AJ0 algorithms_NN2 ._SENT 
There_EX0 are_VBB a_AT0 variety_NN1 of_PRF conic_AJ0 and_CJC ellipse_NN1 fitting_VVG algorithms_NN2 that_CJT have_VHB been_VBN developed_VVN over_PRP the_AT0 last_ORD few_DT0 decades_NN2 ._SENT While_CJS a_AT0 full_AJ0 accounting_NN1 of_PRF these_DT0 various_AJ0 methods_NN2 is_VBZ beyond_PRP the_AT0 scope_NN1 of_PRF the_AT0 present_AJ0 work_NN1 ,_PUN there_EX0 are_VBB both_AV0 non-iterative_AJ0 (_PUL e.g._AV0 ,_PUN [_PUL 60_CRD ]_PUR ,_PUN [_PUL 62_CRD ]_PUR –_PUN [_PUL 64_CRD ]_PUR )_PUR and_CJC iterative_AJ0 methods_NN2 (_PUL e.g._AV0 ,_PUN [_PUL 65_CRD ]_PUR –_PUN [_PUL 67_CRD ]_PUR )_PUR ._SENT In_PRP the_AT0 case_NN1 where_AVQ a_AT0 large_AJ0 number_NN1 of_PRF horizon_NN1 points_NN2 are_VBB observed_VVN over_PRP substantial_AJ0 horizon_NN1 arcs_NN2 ,_PUN we_PNP find_VVB very_AV0 little_DT0 practical_AJ0 difference_NN1 between_PRP the_AT0 best_AJS contemporary_AJ0 non-iterative_AJ0 methods_NN2 (_PUL e.g._AV0 ,_PUN the_AT0 '_POS '_PUQ semihyper_NN1 '_POS '_POS least_DT0 squares_NN2 fit_VVB from_PRP [_PUL 64_CRD ]_PUR and_CJC [_PUL 60_CRD ]_PUR )_PUR and_CJC the_AT0 latest_AJS iterative_AJ0 methods_NN2 ._SENT The_AT0 advantages_NN2 of_PRF the_AT0 iterative_AJ0 methods_NN2 become_VVB more_AV0 apparent_AJ0 when_CJS there_EX0 are_VBB very_AV0 few_DT0 horizon_NN1 points_NN2 or_CJC when_CJS the_AT0 arc_NN1 size_NN1 becomes_VVZ small_AJ0 ._SENT These_DT0 trends_NN2 are_VBB supported_VVN by_PRP extensive_AJ0 numerical_AJ0 experiments_NN2 throughout_PRP the_AT0 remainder_NN1 of_PRF this_DT0 article_NN1 and_CJC are_VBB consistent_AJ0 with_PRP the_AT0 observations_NN2 of_PRF [_PUL 68_CRD ]_PUR ._SENT 
Finally_AV0 ,_PUN note_VVB that_CJT all_DT0 derivations_NN2 in_PRP this_DT0 tutorial_NN1 work_NN1 with_PRP conics_NN2 in_PRP image_NN1 plane_NN1 coordinates_NN2 instead_AV0 of_PRF conics_NN2 in_PRP pixel_NN1 coordinates_NN2 ._SENT The_AT0 distinction_NN1 between_PRP these_DT0 two_CRD is_VBZ discussed_VVN in_PRP Sec._NN1 II-D2_CRD ._SENT Since_PRP horizon_NN1 localization_NN1 algorithms_NN2 usually_AV0 provide_VVB a_AT0 set_NN1 of_PRF pixel_NN1 coordinates_NN2 {_PUL u_ZZ0 ̄i_AJ0 }_PUR ni_NN2 =_NN1 1_CRD ,_PUN there_EX0 are_VBB two_CRD straight-forward_AJ0 ways_NN2 to_TO0 obtain_VVI the_AT0 desired_AJ0 image_NN1 plane_NN1 conic_AJ0 C_ZZ0 :_PUN 
1_CRD )_PUR Transform_VVB {_PUL u_ZZ0 ̄i_AJ0 }_PUR ni_NN2 =_NN1 1_CRD to_PRP {_PUL x_ZZ0 ̄i_AJ0 }_PUR ni_NN2 =_NN1 1_CRD using_VVG Eq._NP0 (_PUL 20_CRD )_PUR ._SENT Apply_VVB a_AT0 conic_AJ0 fitting_AJ0 algorithms_NN2 to_PRP the_AT0 points_NN2 {_PUL x_ZZ0 ̄_NN1 i_ZZ0 }_PUR ni_NN2 =_NN1 1_CRD and_CJC directly_AV0 obtain_VVB C_ZZ0 ._SENT 
2_CRD )_PUR Apply_VVB a_AT0 conic_AJ0 fitting_AJ0 algorithms_NN2 to_PRP the_AT0 points_NN2 {_PUL u_ZZ0 ̄i_AJ0 }_PUR ni_NN2 =_NN1 1_CRD and_CJC directly_AV0 obtain_VVB C_ZZ0 uv_AJ0 ._SENT Transform_VVB C_ZZ0 uv_AJ0 to_PRP C_NP0 using_VVG Eq._NP0 (_PUL 54_CRD )_PUR ._SENT 
Regardless_AV0 of_PRF the_AT0 method_NN1 chosen_VVN ,_PUN we_PNP may_VM0 easily_AV0 find_VVI the_AT0 conic_AJ0 in_PRP image_NN1 plane_NN1 coordinates_NN2 ._SENT From_PRP this_DT0 point_NN1 forward_AV0 we_PNP assume_VVB that_CJT any_DT0 ellipse_NN1 fit_AJ0 to_PRP measured_AJ0 horizon_NN1 points_NN2 may_VM0 be_VBI written_VVN as_CJS either_AV0 C_ZZ0 (_PUL conic_AJ0 locus_NN1 )_PUR or_CJC C∗_NP0 (_PUL conic_AJ0 envelope_NN1 )_PUR ._SENT 
III_CRD ._SENT OPTICAL_AJ0 NAVIGATION_NN1 (_PUL OPNAV_NP0 )_PUR WITH_PRP A_AT0 WELL-KNOWN_AJ0 CELESTIAL_AJ0 BODY_NN1 
The_AT0 first_ORD of_PRF the_AT0 three_CRD motivating_AJ0 scenarios_NN2 from_PRP Sec._NN1 I_PNP is_VBZ the_AT0 case_NN1 where_AVQ the_AT0 relative_AJ0 camera-to-celestial_AJ0 body_NN1 attitude_NN1 is_VBZ known_VVN ,_PUN but_CJC the_AT0 relative_AJ0 position_NN1 is_VBZ unknown_AJ0 ._SENT This_DT0 is_VBZ exactly_AV0 the_AT0 situation_NN1 when_CJS navigating_VVG relative_AJ0 to_PRP well-known_AJ0 celestial_AJ0 bodies_NN2 and_CJC is_VBZ the_AT0 usual_AJ0 assumption_NN1 for_PRP classical_AJ0 OPNAV_NP0 ._SENT The_AT0 objective_NN1 here_AV0 is_VBZ to_TO0 solve_VVI for_PRP the_AT0 unknown_AJ0 location_NN1 of_PRF the_AT0 camera_NN1 (_PUL and_CJC ,_PUN hence_AV0 ,_PUN the_AT0 location_NN1 of_PRF spacecraft_NN0 )_PUR ._SENT 
For_PRP many_DT0 bodies_NN2 of_PRF interest_NN1 (_PUL e.g._AV0 ,_PUN Moon_NP0 ,_PUN Mars_NP0 ,_PUN Earth_NN1 )_PUR we_PNP have_VHB excellent_AJ0 knowledge_NN1 of_PRF both_DT0 shape_NN1 and_CJC orientation_NN1 ._SENT 
