[geometry] Establish Meshcat in C++ (step 3)

Implements the most essential elements of the Meshcat API:
- Added SceneTree data structure.
- Added stduuid (to create the uuid's required by meshcat)
- Added SetObject from Shape, Rgba
  - Supports all shapes that meshcat_visualizer.py supports (more, in fact)
  - Does not support texture maps yet.
- Added SetTransform
- Added Delete
- Added SetProperty<double>
- Added HasPath, GetPacked* methods, primarily for testing.

This PR is > 500 lines, but a significant portion of this commit is relatively boilerplate management of the msgpack data structures used to communicate with the javascript client.

Updated the meshcat_manual_test to step the user through the checks.
Updated meschat_test to hammer on the new API.

geometry/BUILD.bazel

@@ -342,6 +342,18 @@ drake_cc_library(
     ],
 )
 
+drake_cc_library(
+    name = "meshcat_types",
+    hdrs = ["meshcat_types.h"],
+    install_hdrs_exclude = ["meshcat_types.h"],
+    tags = [
+        "exclude_from_libdrake",
+        "exclude_from_package",
+    ],
+    visibility = ["//:__subpackages__"],
+    deps = ["@msgpack"],
+)
+
 drake_cc_library(
     name = "meshcat",
     srcs = ["meshcat.cc"],
@@ -352,10 +364,15 @@ drake_cc_library(
         "@meshcat//:dist/main.min.js",
     ],
     deps = [
+        ":meshcat_types",
+        ":rgba",
+        ":shape_specification",
         "//common:essential",
         "//common:find_resource",
         "//common:unused",
+        "//math:geometric_transform",
         "@msgpack",
+        "@stduuid",
         "@uwebsockets",
     ],
 )
@@ -364,6 +381,7 @@ drake_cc_binary(
     name = "meshcat_manual_test",
     testonly = True,
     srcs = ["test/meshcat_manual_test.cc"],
+    data = ["//systems/sensors:test_models"],
     visibility = ["//visibility:private"],
     deps = [":meshcat"],
 )
@@ -377,11 +395,17 @@ drake_py_binary(
 
 drake_cc_googletest(
     name = "meshcat_test",
-    data = [":meshcat_websocket_client"],
+    data = [
+        ":meshcat_websocket_client",
+        "//systems/sensors:test_models",
+    ],
     # Disable valgrind because it currently fails on the system call to python.
     # TODO(russt): Make it possible to pass --trace-children=no to valgrind.sh.
     tags = ["no_memcheck"],
-    deps = [":meshcat"],
+    deps = [
+        ":meshcat",
+        "//common/test_utilities:eigen_matrix_compare",
+    ],
 )
 
 # -----------------------------------------------------