EPSG Helmert and Molodensky-Badekas methods in the Geographic 2D domain: use the push/pop v_3 operator to preserve the Z component

5 files changed, 179 insertions, 166 deletions

diff --git a/test/cli/tv_out.dist b/test/cli/tv_out.dist
index 72e95634..257e9400 100644
--- a/test/cli/tv_out.dist
+++ b/test/cli/tv_out.dist
@@ -7,8 +7,8 @@ Test NAD27 to raw ellipse
 79d00'00.000"W 35d00'00.000"N 0.0	79dW	35dN 0.000
 ##############################################################
 Between two 3parameter approximations on same ellipsoid
-0d00'00.000"W 0d00'00.000"N 0.0	0dE	0dN 4.000
-79d00'00.000"W 45d00'00.000"N 0.0	78d59'59.821"W	44d59'59.983"N 0.540
+0d00'00.000"W 0d00'00.000"N 0.0	0dE	0dN 0.000
+79d00'00.000"W 45d00'00.000"N 0.0	78d59'59.821"W	44d59'59.983"N 0.000
 ##############################################################
 3param to raw ellipsoid on same ellipsoid
 0d00'00.000"W 0d00'00.000"N 0.0	0dE	0dN 0.000
@@ -352,7 +352,7 @@ Test inverse handling
 10 20	-1384841.19	7581707.88 0.00
 ##############################################################
 Test MGI datum gives expected results (#207)
-16.33 48.20	595710.3731286	5357598.4645652 -44.4951085
+16.33 48.20	595710.3731286	5357598.4645652 0.0000000
 ##############################################################
 Test omerc sensitivity with locations 90d from origin(#114)
 56.958381652832 72.8798	-9985.16336453	-227.67701050 0.00000000
@@ -387,10 +387,10 @@ Test bug 244 (2)
 987122.418330284 -14429896.539530909	-140.10000000000	-87.00000000000 0.00000000000
 ##############################################################
 Test bug 245 (use +datum=carthage)
-10 34	592302.9819461	3762148.7340609 -30.3110170
+10 34	592302.9819461	3762148.7340609 0.0000000
 ##############################################################
 Test bug 245 (use expansion of +datum=carthage)
-10 34	592302.9819461	3762148.7340609 -30.3110170
+10 34	592302.9819461	3762148.7340609 0.0000000
 ##############################################################
 Test SCH forward projection
 0.0 0.0	-1977112.0305592	5551475.1418378 6595.7256583
diff --git a/test/unit/gie_self_tests.cpp b/test/unit/gie_self_tests.cpp
index b77fecf3..a3cd8eab 100644
--- a/test/unit/gie_self_tests.cpp
+++ b/test/unit/gie_self_tests.cpp
@@ -682,17 +682,17 @@ TEST(gie, proj_create_crs_to_crs_PULKOVO42_ETRS89) {
     EXPECT_NEAR(c.xy.x, 44.999701238, 1e-9);
     EXPECT_NEAR(c.xy.y, 24.998474948, 1e-9);
     EXPECT_EQ(std::string(proj_pj_info(P).definition),
-              "proj=pipeline step proj=axisswap order=2,1 step "
-              "proj=unitconvert xy_in=deg xy_out=rad step proj=cart "
+              "proj=pipeline step proj=push v_3 step proj=axisswap order=2,1 "
+              "step proj=unitconvert xy_in=deg xy_out=rad step proj=cart "
               "ellps=krass step proj=helmert x=2.3287 y=-147.0425 z=-92.0802 "
               "rx=0.3092483 ry=-0.32482185 rz=-0.49729934 s=5.68906266 "
               "convention=coordinate_frame step inv proj=cart ellps=GRS80 step "
               "proj=unitconvert xy_in=rad xy_out=deg step proj=axisswap "
-              "order=2,1");
+              "order=2,1 step proj=pop v_3");
 
     c = proj_trans(P, PJ_INV, c);
-    EXPECT_NEAR(c.xy.x, 45, 1e-9);
-    EXPECT_NEAR(c.xy.y, 25, 1e-9);
+    EXPECT_NEAR(c.xy.x, 45, 1e-8);
+    EXPECT_NEAR(c.xy.y, 25, 1e-8);
 
     c.xyz.x = 45; // Lat
     c.xyz.y = 25; // Long
@@ -711,12 +711,12 @@ TEST(gie, proj_create_crs_to_crs_PULKOVO42_ETRS89) {
     EXPECT_NEAR(c.xy.x, 51.999714150, 1e-9);
     EXPECT_NEAR(c.xy.y, 19.998187811, 1e-9);
     EXPECT_EQ(std::string(proj_pj_info(P).definition),
-              "proj=pipeline step proj=axisswap order=2,1 step "
-              "proj=unitconvert xy_in=deg xy_out=rad step proj=cart "
+              "proj=pipeline step proj=push v_3 step proj=axisswap order=2,1 "
+              "step proj=unitconvert xy_in=deg xy_out=rad step proj=cart "
               "ellps=krass step proj=helmert x=33.4 y=-146.6 z=-76.3 rx=-0.359 "
               "ry=-0.053 rz=0.844 s=-0.84 convention=position_vector step inv "
               "proj=cart ellps=GRS80 step proj=unitconvert xy_in=rad "
-              "xy_out=deg step proj=axisswap order=2,1");
+              "xy_out=deg step proj=axisswap order=2,1 step proj=pop v_3");
 
     proj_destroy(P);
 }
diff --git a/test/unit/test_c_api.cpp b/test/unit/test_c_api.cpp
index d535e412..273a04e6 100644
--- a/test/unit/test_c_api.cpp
+++ b/test/unit/test_c_api.cpp
@@ -838,13 +838,14 @@ TEST_F(CApi, proj_create_from_database) {
         ASSERT_NE(info.definition, nullptr);
         EXPECT_EQ(
             info.definition,
-            std::string("proj=pipeline step proj=axisswap order=2,1 step "
-                        "proj=unitconvert xy_in=deg xy_out=rad step proj=cart "
-                        "ellps=bessel step proj=helmert x=601.705 y=84.263 "
-                        "z=485.227 rx=-4.7354 ry=-1.3145 rz=-5.393 s=-2.3887 "
-                        "convention=coordinate_frame step inv proj=cart "
-                        "ellps=GRS80 step proj=unitconvert xy_in=rad "
-                        "xy_out=deg step proj=axisswap order=2,1"));
+            std::string("proj=pipeline step proj=push v_3 step proj=axisswap "
+                        "order=2,1 step proj=unitconvert xy_in=deg xy_out=rad "
+                        "step proj=cart ellps=bessel step proj=helmert "
+                        "x=601.705 y=84.263 z=485.227 rx=-4.7354 ry=-1.3145 "
+                        "rz=-5.393 s=-2.3887 convention=coordinate_frame step "
+                        "inv proj=cart ellps=GRS80 step proj=unitconvert "
+                        "xy_in=rad xy_out=deg step proj=axisswap order=2,1 "
+                        "step proj=pop v_3"));
         EXPECT_EQ(info.accuracy, 1);
     }
 }
diff --git a/test/unit/test_factory.cpp b/test/unit/test_factory.cpp
index ce019079..64512e56 100644
--- a/test/unit/test_factory.cpp
+++ b/test/unit/test_factory.cpp
@@ -658,13 +658,13 @@ TEST(factory, AuthorityFactory_createCoordinateOperation_helmert_3) {
                  NoSuchAuthorityCodeException);
     auto op = factory->createCoordinateOperation("1113", false);
     EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
-              "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-              "+proj=unitconvert +xy_in=deg +xy_out=rad +step +inv "
-              "+proj=longlat +a=6378249.145 +rf=293.4663077 +step +proj=cart "
-              "+a=6378249.145 +rf=293.4663077 +step +proj=helmert +x=-143 "
-              "+y=-90 +z=-294 +step +inv +proj=cart +ellps=WGS84 +step "
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=deg +xy_out=rad +step "
+              "+inv +proj=longlat +a=6378249.145 +rf=293.4663077 +step "
+              "+proj=cart +a=6378249.145 +rf=293.4663077 +step +proj=helmert "
+              "+x=-143 +y=-90 +z=-294 +step +inv +proj=cart +ellps=WGS84 +step "
               "+proj=unitconvert +xy_in=rad +xy_out=deg +step +proj=axisswap "
-              "+order=2,1");
+              "+order=2,1 +step +proj=pop +v_3");
 }
 
 // ---------------------------------------------------------------------------
@@ -673,13 +673,13 @@ TEST(factory, AuthorityFactory_createCoordinateOperation_helmert_7_CF) {
     auto factory = AuthorityFactory::create(DatabaseContext::create(), "EPSG");
     auto op = factory->createCoordinateOperation("7676", false);
     EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
-              "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-              "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-              "+ellps=bessel +step +proj=helmert +x=577.88891 +y=165.22205 "
-              "+z=391.18289 +rx=-4.9145 +ry=0.94729 +rz=13.05098 +s=7.78664 "
-              "+convention=coordinate_frame +step +inv +proj=cart +ellps=WGS84 "
-              "+step +proj=unitconvert +xy_in=rad +xy_out=deg +step "
-              "+proj=axisswap +order=2,1");
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=deg +xy_out=rad +step "
+              "+proj=cart +ellps=bessel +step +proj=helmert +x=577.88891 "
+              "+y=165.22205 +z=391.18289 +rx=-4.9145 +ry=0.94729 +rz=13.05098 "
+              "+s=7.78664 +convention=coordinate_frame +step +inv +proj=cart "
+              "+ellps=WGS84 +step +proj=unitconvert +xy_in=rad +xy_out=deg "
+              "+step +proj=axisswap +order=2,1 +step +proj=pop +v_3");
 }
 
 // ---------------------------------------------------------------------------
@@ -850,14 +850,15 @@ TEST(factory,
     EXPECT_TRUE(so->validateParameters().empty());
 
     EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
-              "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-              "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-              "+ellps=bessel +step +proj=molobadekas +x=593.032 +y=26 "
-              "+z=478.741 +rx=0.409394387439237 +ry=-0.359705195614311 "
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=deg +xy_out=rad +step "
+              "+proj=cart +ellps=bessel +step +proj=molobadekas +x=593.032 "
+              "+y=26 +z=478.741 +rx=0.409394387439237 +ry=-0.359705195614311 "
               "+rz=1.86849100035057 +s=4.0772 +px=3903453.148 +py=368135.313 "
               "+pz=5012970.306 +convention=coordinate_frame +step +inv "
               "+proj=cart +ellps=GRS80 +step +proj=unitconvert +xy_in=rad "
-              "+xy_out=deg +step +proj=axisswap +order=2,1");
+              "+xy_out=deg +step +proj=axisswap +order=2,1 +step +proj=pop "
+              "+v_3");
 }
 
 // ---------------------------------------------------------------------------
@@ -2065,11 +2066,12 @@ TEST_F(FactoryWithTmpDatabase, AuthorityFactory_wkt_based_transformation) {
     ASSERT_EQ(res.size(), 1);
     EXPECT_EQ(res[0]->nameStr(), "My WKT string based op");
     EXPECT_EQ(res[0]->exportToPROJString(PROJStringFormatter::create().get()),
-              "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-              "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-              "+ellps=WGS84 +step +proj=helmert +x=1 +y=2 +z=3 +step +inv "
-              "+proj=cart +ellps=WGS84 +step +proj=unitconvert +xy_in=rad "
-              "+xy_out=deg +step +proj=axisswap +order=2,1");
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=deg +xy_out=rad +step "
+              "+proj=cart +ellps=WGS84 +step +proj=helmert +x=1 +y=2 +z=3 "
+              "+step +inv +proj=cart +ellps=WGS84 +step +proj=unitconvert "
+              "+xy_in=rad +xy_out=deg +step +proj=axisswap +order=2,1 +step "
+              "+proj=pop +v_3");
 }
 
 // ---------------------------------------------------------------------------
diff --git a/test/unit/test_operation.cpp b/test/unit/test_operation.cpp
index 6144663e..e855a818 100644
--- a/test/unit/test_operation.cpp
+++ b/test/unit/test_operation.cpp
@@ -561,11 +561,12 @@ TEST(operation, transformation_createGeocentricTranslations) {
     EXPECT_EQ(inv_transf_as_transf->getTOWGS84Parameters(), expected_inv);
 
     EXPECT_EQ(transf->exportToPROJString(PROJStringFormatter::create().get()),
-              "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-              "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-              "+ellps=GRS80 +step +proj=helmert +x=1 +y=2 +z=3 +step +inv "
-              "+proj=cart +ellps=WGS84 +step +proj=unitconvert +xy_in=rad "
-              "+xy_out=deg +step +proj=axisswap +order=2,1");
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=deg +xy_out=rad +step "
+              "+proj=cart +ellps=GRS80 +step +proj=helmert +x=1 +y=2 +z=3 "
+              "+step +inv +proj=cart +ellps=WGS84 +step +proj=unitconvert "
+              "+xy_in=rad +xy_out=deg +step +proj=axisswap +order=2,1 +step "
+              "+proj=pop +v_3");
 }
 
 // ---------------------------------------------------------------------------
@@ -668,13 +669,13 @@ TEST(operation, transformation_createPositionVector) {
     EXPECT_EQ(transf->getTOWGS84Parameters(), expected);
 
     EXPECT_EQ(transf->exportToPROJString(PROJStringFormatter::create().get()),
-              "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-              "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-              "+ellps=GRS80 +step +proj=helmert +x=1 +y=2 +z=3 +rx=4 +ry=5 "
-              "+rz=6 +s=7 +convention=position_vector +step +inv +proj=cart "
-              "+ellps=WGS84 +step "
-              "+proj=unitconvert +xy_in=rad +xy_out=deg +step +proj=axisswap "
-              "+order=2,1");
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=deg +xy_out=rad +step "
+              "+proj=cart +ellps=GRS80 +step +proj=helmert +x=1 +y=2 +z=3 "
+              "+rx=4 +ry=5 +rz=6 +s=7 +convention=position_vector +step +inv "
+              "+proj=cart +ellps=WGS84 +step +proj=unitconvert +xy_in=rad "
+              "+xy_out=deg +step +proj=axisswap +order=2,1 +step +proj=pop "
+              "+v_3");
 
     auto inv_transf = transf->inverse();
     ASSERT_EQ(inv_transf->coordinateOperationAccuracies().size(), 1);
@@ -691,12 +692,12 @@ TEST(operation, transformation_createPositionVector) {
 #else
     EXPECT_EQ(
         inv_transf->exportToPROJString(PROJStringFormatter::create().get()),
-        "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-        "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-        "+ellps=WGS84 +step +inv +proj=helmert +x=1 +y=2 +z=3 +rx=4 "
-        "+ry=5 +rz=6 +s=7 +convention=position_vector +step +inv "
-        "+proj=cart +ellps=GRS80 +step +proj=unitconvert +xy_in=rad "
-        "+xy_out=deg +step +proj=axisswap +order=2,1");
+        "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap +order=2,1 "
+        "+step +proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
+        "+ellps=WGS84 +step +inv +proj=helmert +x=1 +y=2 +z=3 +rx=4 +ry=5 "
+        "+rz=6 +s=7 +convention=position_vector +step +inv +proj=cart "
+        "+ellps=GRS80 +step +proj=unitconvert +xy_in=rad +xy_out=deg +step "
+        "+proj=axisswap +order=2,1 +step +proj=pop +v_3");
 
     // In WKT, use approximate formula
     auto wkt = inv_transf->exportToWKT(WKTFormatter::create().get());
@@ -737,13 +738,13 @@ TEST(operation, transformation_createCoordinateFrameRotation) {
     EXPECT_EQ(params, expected);
 
     EXPECT_EQ(transf->exportToPROJString(PROJStringFormatter::create().get()),
-              "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-              "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-              "+ellps=GRS80 +step +proj=helmert +x=1 +y=2 +z=3 +rx=-4 +ry=-5 "
-              "+rz=-6 +s=7 +convention=coordinate_frame +step +inv +proj=cart "
-              "+ellps=WGS84 +step "
-              "+proj=unitconvert +xy_in=rad +xy_out=deg +step +proj=axisswap "
-              "+order=2,1");
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=deg +xy_out=rad +step "
+              "+proj=cart +ellps=GRS80 +step +proj=helmert +x=1 +y=2 +z=3 "
+              "+rx=-4 +ry=-5 +rz=-6 +s=7 +convention=coordinate_frame +step "
+              "+inv +proj=cart +ellps=WGS84 +step +proj=unitconvert +xy_in=rad "
+              "+xy_out=deg +step +proj=axisswap +order=2,1 +step +proj=pop "
+              "+v_3");
 
     auto inv_transf = transf->inverse();
     ASSERT_EQ(inv_transf->coordinateOperationAccuracies().size(), 0);
@@ -760,12 +761,12 @@ TEST(operation, transformation_createCoordinateFrameRotation) {
 #else
     EXPECT_EQ(
         inv_transf->exportToPROJString(PROJStringFormatter::create().get()),
-        "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-        "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-        "+ellps=WGS84 +step +inv +proj=helmert +x=1 +y=2 +z=3 +rx=-4 "
-        "+ry=-5 +rz=-6 +s=7 +convention=coordinate_frame +step +inv "
-        "+proj=cart +ellps=GRS80 +step +proj=unitconvert +xy_in=rad "
-        "+xy_out=deg +step +proj=axisswap +order=2,1");
+        "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap +order=2,1 "
+        "+step +proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
+        "+ellps=WGS84 +step +inv +proj=helmert +x=1 +y=2 +z=3 +rx=-4 +ry=-5 "
+        "+rz=-6 +s=7 +convention=coordinate_frame +step +inv +proj=cart "
+        "+ellps=GRS80 +step +proj=unitconvert +xy_in=rad +xy_out=deg +step "
+        "+proj=axisswap +order=2,1 +step +proj=pop +v_3");
 
     // In WKT, use approximate formula
     auto wkt = inv_transf->exportToWKT(WKTFormatter::create().get());
@@ -4197,13 +4198,13 @@ TEST(operation, geogCRS_to_geogCRS_context_default) {
 
         EXPECT_EQ(
             list[0]->exportToPROJString(PROJStringFormatter::create().get()),
-            "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-            "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-            "+ellps=krass +step +proj=helmert +x=2.3287 +y=-147.0425 "
-            "+z=-92.0802 +rx=0.3092483 +ry=-0.32482185 +rz=-0.49729934 "
-            "+s=5.68906266 +convention=coordinate_frame +step +inv "
-            "+proj=cart +ellps=GRS80 +step +proj=unitconvert +xy_in=rad "
-            "+xy_out=deg +step +proj=axisswap +order=2,1");
+            "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+            "+order=2,1 +step +proj=unitconvert +xy_in=deg +xy_out=rad +step "
+            "+proj=cart +ellps=krass +step +proj=helmert +x=2.3287 "
+            "+y=-147.0425 +z=-92.0802 +rx=0.3092483 +ry=-0.32482185 "
+            "+rz=-0.49729934 +s=5.68906266 +convention=coordinate_frame +step "
+            "+inv +proj=cart +ellps=GRS80 +step +proj=unitconvert +xy_in=rad "
+            "+xy_out=deg +step +proj=axisswap +order=2,1 +step +proj=pop +v_3");
     }
 
     // Reverse case
@@ -4218,13 +4219,13 @@ TEST(operation, geogCRS_to_geogCRS_context_default) {
 
         EXPECT_EQ(
             list[0]->exportToPROJString(PROJStringFormatter::create().get()),
-            "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-            "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-            "+ellps=GRS80 +step +inv +proj=helmert +x=2.3287 "
+            "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+            "+order=2,1 +step +proj=unitconvert +xy_in=deg +xy_out=rad +step "
+            "+proj=cart +ellps=GRS80 +step +inv +proj=helmert +x=2.3287 "
             "+y=-147.0425 +z=-92.0802 +rx=0.3092483 +ry=-0.32482185 "
-            "+rz=-0.49729934 +s=5.68906266 +convention=coordinate_frame "
-            "+step +inv +proj=cart +ellps=krass +step +proj=unitconvert "
-            "+xy_in=rad +xy_out=deg +step +proj=axisswap +order=2,1");
+            "+rz=-0.49729934 +s=5.68906266 +convention=coordinate_frame +step "
+            "+inv +proj=cart +ellps=krass +step +proj=unitconvert +xy_in=rad "
+            "+xy_out=deg +step +proj=axisswap +order=2,1 +step +proj=pop +v_3");
     }
 }
 
@@ -4358,11 +4359,12 @@ TEST(operation, geogCRS_to_geogCRS_context_inverse_needed) {
         ASSERT_EQ(list.size(), 3);
         EXPECT_EQ(
             list[0]->exportToPROJString(PROJStringFormatter::create().get()),
-            "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-            "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-            "+ellps=clrk80ign +step +proj=helmert +x=-168 +y=-60 +z=320 "
-            "+step +inv +proj=cart +ellps=GRS80 +step +proj=unitconvert "
-            "+xy_in=rad +xy_out=deg +step +proj=axisswap +order=2,1");
+            "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+            "+order=2,1 +step +proj=unitconvert +xy_in=deg +xy_out=rad +step "
+            "+proj=cart +ellps=clrk80ign +step +proj=helmert +x=-168 +y=-60 "
+            "+z=320 +step +inv +proj=cart +ellps=GRS80 +step +proj=unitconvert "
+            "+xy_in=rad +xy_out=deg +step +proj=axisswap +order=2,1 +step "
+            "+proj=pop +v_3");
         EXPECT_EQ(list[1]->exportToPROJString(
                       PROJStringFormatter::create(
                           PROJStringFormatter::Convention::PROJ_5,
@@ -5299,15 +5301,15 @@ TEST(operation, boundCRS_of_geogCRS_to_geogCRS) {
     auto op = CoordinateOperationFactory::create()->createOperation(
         boundCRS, GeographicCRS::EPSG_4326);
     ASSERT_TRUE(op != nullptr);
-    EXPECT_EQ(
-        op->exportToPROJString(PROJStringFormatter::create().get()),
-        "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-        "+proj=unitconvert +xy_in=grad +xy_out=rad +step +inv +proj=longlat "
-        "+ellps=clrk80ign +pm=paris +step +proj=cart +ellps=clrk80ign "
-        "+step +proj=helmert +x=1 +y=2 +z=3 +rx=4 +ry=5 +rz=6 +s=7 "
-        "+convention=position_vector +step "
-        "+inv +proj=cart +ellps=WGS84 +step +proj=unitconvert +xy_in=rad "
-        "+xy_out=deg +step +proj=axisswap +order=2,1");
+    EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=grad +xy_out=rad "
+              "+step +inv +proj=longlat +ellps=clrk80ign +pm=paris +step "
+              "+proj=cart +ellps=clrk80ign +step +proj=helmert +x=1 +y=2 +z=3 "
+              "+rx=4 +ry=5 +rz=6 +s=7 +convention=position_vector +step +inv "
+              "+proj=cart +ellps=WGS84 +step +proj=unitconvert +xy_in=rad "
+              "+xy_out=deg +step +proj=axisswap +order=2,1 +step +proj=pop "
+              "+v_3");
 }
 
 // ---------------------------------------------------------------------------
@@ -5321,12 +5323,13 @@ TEST(operation, boundCRS_of_geogCRS_to_geogCRS_with_area) {
         boundCRS, authFactory->createCoordinateReferenceSystem("4326"));
     ASSERT_TRUE(op != nullptr);
     EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
-              "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-              "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-              "+ellps=clrk66 +step +proj=helmert +x=1 +y=2 +z=3 +rx=4 +ry=5 "
-              "+rz=6 +s=7 +convention=position_vector +step +inv +proj=cart "
-              "+ellps=WGS84 +step +proj=unitconvert +xy_in=rad +xy_out=deg "
-              "+step +proj=axisswap +order=2,1");
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=deg +xy_out=rad +step "
+              "+proj=cart +ellps=clrk66 +step +proj=helmert +x=1 +y=2 +z=3 "
+              "+rx=4 +ry=5 +rz=6 +s=7 +convention=position_vector +step +inv "
+              "+proj=cart +ellps=WGS84 +step +proj=unitconvert +xy_in=rad "
+              "+xy_out=deg +step +proj=axisswap +order=2,1 +step +proj=pop "
+              "+v_3");
 }
 
 // ---------------------------------------------------------------------------
@@ -5362,11 +5365,11 @@ TEST(operation, createOperation_boundCRS_identified_by_datum) {
         NN_CHECK_ASSERT(src), NN_CHECK_ASSERT(dest));
     ASSERT_TRUE(op != nullptr);
     EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
-              "+proj=pipeline +step +proj=unitconvert +xy_in=deg +xy_out=rad "
-              "+step +proj=cart +ellps=WGS84 "
-              "+step +proj=helmert +x=263 +y=-6 +z=-431 "
-              "+step +inv +proj=cart +ellps=clrk80ign "
-              "+step +proj=utm +zone=32 +ellps=clrk80ign");
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=unitconvert "
+              "+xy_in=deg +xy_out=rad +step +proj=cart +ellps=WGS84 +step "
+              "+proj=helmert +x=263 +y=-6 +z=-431 +step +inv +proj=cart "
+              "+ellps=clrk80ign +step +proj=pop +v_3 +step +proj=utm +zone=32 "
+              "+ellps=clrk80ign");
 
     auto authFactory =
         AuthorityFactory::create(DatabaseContext::create(), std::string());
@@ -5436,12 +5439,13 @@ TEST(operation, boundCRS_of_projCRS_to_geogCRS) {
     ASSERT_TRUE(op != nullptr);
     EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
               "+proj=pipeline +step +inv +proj=utm +zone=31 +ellps=clrk80ign "
-              "+pm=paris "
-              "+step +proj=cart +ellps=clrk80ign +step +proj=helmert +x=1 +y=2 "
-              "+z=3 +rx=4 +ry=5 +rz=6 +s=7 +convention=position_vector +step "
-              "+inv +proj=cart +ellps=WGS84 "
+              "+pm=paris +step +proj=longlat +ellps=clrk80ign +pm=paris +step "
+              "+proj=push +v_3 +step +inv +proj=longlat +ellps=clrk80ign "
+              "+pm=paris +step +proj=cart +ellps=clrk80ign +step +proj=helmert "
+              "+x=1 +y=2 +z=3 +rx=4 +ry=5 +rz=6 +s=7 "
+              "+convention=position_vector +step +inv +proj=cart +ellps=WGS84 "
               "+step +proj=unitconvert +xy_in=rad +xy_out=deg +step "
-              "+proj=axisswap +order=2,1");
+              "+proj=axisswap +order=2,1 +step +proj=pop +v_3");
 }
 
 // ---------------------------------------------------------------------------
@@ -5456,15 +5460,14 @@ TEST(operation, boundCRS_of_geogCRS_to_projCRS) {
     auto op =
         CoordinateOperationFactory::create()->createOperation(boundCRS, utm31);
     ASSERT_TRUE(op != nullptr);
-    EXPECT_EQ(
-        op->exportToPROJString(PROJStringFormatter::create().get()),
-        "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-        "+proj=unitconvert +xy_in=grad +xy_out=rad +step +inv +proj=longlat "
-        "+ellps=clrk80ign +pm=paris +step +proj=cart +ellps=clrk80ign "
-        "+step +proj=helmert +x=1 +y=2 +z=3 +rx=4 +ry=5 +rz=6 +s=7 "
-        "+convention=position_vector +step "
-        "+inv +proj=cart +ellps=WGS84 +step +proj=utm +zone=31 "
-        "+ellps=WGS84");
+    EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=grad +xy_out=rad "
+              "+step +inv +proj=longlat +ellps=clrk80ign +pm=paris +step "
+              "+proj=cart +ellps=clrk80ign +step +proj=helmert +x=1 +y=2 +z=3 "
+              "+rx=4 +ry=5 +rz=6 +s=7 +convention=position_vector +step +inv "
+              "+proj=cart +ellps=WGS84 +step +proj=pop +v_3 +step +proj=utm "
+              "+zone=31 +ellps=WGS84");
 }
 
 // ---------------------------------------------------------------------------
@@ -5476,13 +5479,14 @@ TEST(operation, geogCRS_to_boundCRS_of_geogCRS) {
         GeographicCRS::EPSG_4326, boundCRS);
     ASSERT_TRUE(op != nullptr);
     EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
-              "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-              "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-              "+ellps=WGS84 +step +inv +proj=helmert +x=1 +y=2 +z=3 +rx=4 "
-              "+ry=5 +rz=6 +s=7 +convention=position_vector +step +inv "
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=deg +xy_out=rad +step "
+              "+proj=cart +ellps=WGS84 +step +inv +proj=helmert +x=1 +y=2 +z=3 "
+              "+rx=4 +ry=5 +rz=6 +s=7 +convention=position_vector +step +inv "
               "+proj=cart +ellps=clrk80ign +step +proj=longlat "
               "+ellps=clrk80ign +pm=paris +step +proj=unitconvert +xy_in=rad "
-              "+xy_out=grad +step +proj=axisswap +order=2,1");
+              "+xy_out=grad +step +proj=axisswap +order=2,1 +step +proj=pop "
+              "+v_3");
 }
 
 // ---------------------------------------------------------------------------
@@ -5505,12 +5509,14 @@ TEST(operation, boundCRS_to_boundCRS) {
     ASSERT_TRUE(op != nullptr);
     EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
               "+proj=pipeline +step +inv +proj=utm +zone=31 +ellps=clrk80ign "
-              "+pm=paris "
-              "+step +proj=cart +ellps=clrk80ign +step +proj=helmert +x=1 +y=2 "
-              "+z=3 +rx=4 +ry=5 +rz=6 +s=7 +convention=position_vector +step "
-              "+inv +proj=helmert +x=8 +y=9 +z=10 +rx=11 +ry=12 +rz=13 +s=14 "
-              "+convention=position_vector +step +inv +proj=cart +ellps=GRS80 "
-              "+step +proj=utm +zone=32 +ellps=GRS80");
+              "+pm=paris +step +proj=longlat +ellps=clrk80ign +pm=paris +step "
+              "+proj=push +v_3 +step +inv +proj=longlat +ellps=clrk80ign "
+              "+pm=paris +step +proj=cart +ellps=clrk80ign +step +proj=helmert "
+              "+x=1 +y=2 +z=3 +rx=4 +ry=5 +rz=6 +s=7 "
+              "+convention=position_vector +step +inv +proj=helmert +x=8 +y=9 "
+              "+z=10 +rx=11 +ry=12 +rz=13 +s=14 +convention=position_vector "
+              "+step +inv +proj=cart +ellps=GRS80 +step +proj=pop +v_3 +step "
+              "+proj=utm +zone=32 +ellps=GRS80");
 }
 
 // ---------------------------------------------------------------------------
@@ -5523,13 +5529,13 @@ TEST(operation, boundCRS_to_boundCRS_noop_for_TOWGS84) {
     auto op = CoordinateOperationFactory::create()->createOperation(boundCRS1,
                                                                     boundCRS2);
     ASSERT_TRUE(op != nullptr);
-    EXPECT_EQ(
-        op->exportToPROJString(PROJStringFormatter::create().get()),
-        "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-        "+proj=unitconvert +xy_in=grad +xy_out=rad +step +inv +proj=longlat "
-        "+ellps=clrk80ign +pm=paris +step +proj=cart +ellps=clrk80ign "
-        "+step +inv +proj=cart +ellps=GRS80 +step +proj=unitconvert "
-        "+xy_in=rad +xy_out=deg +step +proj=axisswap +order=2,1");
+    EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=grad +xy_out=rad "
+              "+step +inv +proj=longlat +ellps=clrk80ign +pm=paris +step "
+              "+proj=cart +ellps=clrk80ign +step +inv +proj=cart +ellps=GRS80 "
+              "+step +proj=unitconvert +xy_in=rad +xy_out=deg +step "
+              "+proj=axisswap +order=2,1 +step +proj=pop +v_3");
 }
 
 // ---------------------------------------------------------------------------
@@ -5927,12 +5933,13 @@ TEST(operation, compoundCRS_with_boundGeogCRS_to_geogCRS) {
         compound, GeographicCRS::EPSG_4979);
     ASSERT_TRUE(op != nullptr);
     EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
-              "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-              "+proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart "
-              "+ellps=WGS84 +step +proj=helmert +x=1 +y=2 +z=3 +rx=4 +ry=5 "
-              "+rz=6 +s=7 +convention=position_vector +step +inv +proj=cart "
-              "+ellps=WGS84 +step +proj=unitconvert +xy_in=rad +xy_out=deg "
-              "+step +proj=axisswap +order=2,1");
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=deg +xy_out=rad +step "
+              "+proj=cart +ellps=WGS84 +step +proj=helmert +x=1 +y=2 +z=3 "
+              "+rx=4 +ry=5 +rz=6 +s=7 +convention=position_vector +step +inv "
+              "+proj=cart +ellps=WGS84 +step +proj=unitconvert +xy_in=rad "
+              "+xy_out=deg +step +proj=axisswap +order=2,1 +step +proj=pop "
+              "+v_3");
 }
 
 // ---------------------------------------------------------------------------
@@ -5950,15 +5957,15 @@ TEST(operation, compoundCRS_with_boundGeogCRS_and_boundVerticalCRS_to_geogCRS) {
     // Not completely sure the order of horizontal and vertical operations
     // makes sense
     EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
-              "+proj=pipeline +step +proj=axisswap +order=2,1 +step "
-              "+proj=unitconvert +xy_in=grad +xy_out=rad +step +inv "
-              "+proj=longlat +ellps=clrk80ign +pm=paris +step +proj=cart "
-              "+ellps=clrk80ign +step +proj=helmert +x=1 +y=2 +z=3 +rx=4 +ry=5 "
-              "+rz=6 +s=7 +convention=position_vector +step +inv +proj=cart "
-              "+ellps=WGS84 +step +proj=vgridshift +grids=egm08_25.gtx "
-              "+multiplier=1 +step "
-              "+proj=unitconvert +xy_in=rad +xy_out=deg +step "
-              "+proj=axisswap +order=2,1");
+              "+proj=pipeline +step +proj=push +v_3 +step +proj=axisswap "
+              "+order=2,1 +step +proj=unitconvert +xy_in=grad +xy_out=rad "
+              "+step +inv +proj=longlat +ellps=clrk80ign +pm=paris +step "
+              "+proj=cart +ellps=clrk80ign +step +proj=helmert +x=1 +y=2 +z=3 "
+              "+rx=4 +ry=5 +rz=6 +s=7 +convention=position_vector +step +inv "
+              "+proj=cart +ellps=WGS84 +step +proj=pop +v_3 +step "
+              "+proj=vgridshift +grids=egm08_25.gtx +multiplier=1 +step "
+              "+proj=unitconvert +xy_in=rad +xy_out=deg +step +proj=axisswap "
+              "+order=2,1");
 
     auto grids = op->gridsNeeded(DatabaseContext::create());
     EXPECT_EQ(grids.size(), 1);
@@ -5989,12 +5996,14 @@ TEST(operation, compoundCRS_with_boundProjCRS_and_boundVerticalCRS_to_geogCRS) {
     // makes sense
     EXPECT_EQ(op->exportToPROJString(PROJStringFormatter::create().get()),
               "+proj=pipeline +step +inv +proj=utm +zone=31 +ellps=clrk80ign "
+              "+pm=paris +step +proj=longlat +ellps=clrk80ign +pm=paris +step "
+              "+proj=push +v_3 +step +inv +proj=longlat +ellps=clrk80ign "
               "+pm=paris +step +proj=cart +ellps=clrk80ign +step +proj=helmert "
               "+x=1 +y=2 +z=3 +rx=4 +ry=5 +rz=6 +s=7 "
               "+convention=position_vector +step +inv +proj=cart +ellps=WGS84 "
-              "+step +proj=vgridshift +grids=egm08_25.gtx +multiplier=1 +step "
-              "+proj=unitconvert +xy_in=rad +xy_out=deg +step "
-              "+proj=axisswap +order=2,1");
+              "+step +proj=pop +v_3 +step +proj=vgridshift +grids=egm08_25.gtx "
+              "+multiplier=1 +step +proj=unitconvert +xy_in=rad +xy_out=deg "
+              "+step +proj=axisswap +order=2,1");
 
     auto opInverse = CoordinateOperationFactory::create()->createOperation(
         GeographicCRS::EPSG_4979, compound);
@@ -6200,10 +6209,11 @@ TEST(operation, IGNF_LAMB1_TO_EPSG_4326) {
     EXPECT_EQ(list[1]->exportToPROJString(PROJStringFormatter::create().get()),
               "+proj=pipeline +step +inv +proj=lcc +lat_1=49.5 +lat_0=49.5 "
               "+lon_0=0 +k_0=0.99987734 +x_0=600000 +y_0=200000 "
-              "+ellps=clrk80ign +pm=paris +step +proj=cart +ellps=clrk80ign "
-              "+step +proj=helmert +x=-168 +y=-60 +z=320 +step +inv +proj=cart "
-              "+ellps=WGS84 +step +proj=unitconvert +xy_in=rad +xy_out=deg "
-              "+step +proj=axisswap +order=2,1");
+              "+ellps=clrk80ign +pm=paris +step +proj=push +v_3 +step "
+              "+proj=cart +ellps=clrk80ign +step +proj=helmert +x=-168 +y=-60 "
+              "+z=320 +step +inv +proj=cart +ellps=WGS84 +step "
+              "+proj=unitconvert +xy_in=rad +xy_out=deg +step +proj=axisswap "
+              "+order=2,1 +step +proj=pop +v_3");
 
     auto list2 = CoordinateOperationFactory::create()->createOperations(
         AuthorityFactory::create(DatabaseContext::create(), "EPSG")